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Theorem vdwlem6 15690

Description: Lemma for vdw 15698. (Contributed by Mario Carneiro, 13-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem3.v	⊢ (𝜑 → 𝑉 ∈ ℕ)
vdwlem3.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem4.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem4.h	⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem4.f	⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
vdwlem7.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
vdwlem7.g	⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
vdwlem7.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem7.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
vdwlem7.d	⊢ (𝜑 → 𝐷 ∈ ℕ)
vdwlem7.s	⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
vdwlem6.b	⊢ (𝜑 → 𝐵 ∈ ℕ)
vdwlem6.e	⊢ (𝜑 → 𝐸:(1...𝑀)⟶ℕ)
vdwlem6.s	⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
vdwlem6.j	⊢ 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸‘𝑖))))
vdwlem6.r	⊢ (𝜑 → (#‘ran 𝐽) = 𝑀)
vdwlem6.t	⊢ 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
vdwlem6.p	⊢ 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)))

**Assertion**
Ref	Expression
vdwlem6	⊢ (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑖,𝑗,𝑥,𝑦,𝐺 𝑖,𝐾,𝑗,𝑥,𝑦 𝑖,𝐽,𝑗,𝑥 𝑃,𝑖,𝑥 𝜑,𝑖,𝑗,𝑥,𝑦 𝑅,𝑖,𝑥,𝑦 𝐵,𝑖,𝑗,𝑥,𝑦 𝑖,𝐻,𝑥,𝑦 𝑖,𝑀,𝑗,𝑥,𝑦 𝐷,𝑗,𝑥,𝑦 𝑖,𝐸,𝑗,𝑥,𝑦 𝑖,𝑊,𝑗,𝑥,𝑦 𝑇,𝑖,𝑥 𝑥,𝑉,𝑦 Allowed substitution hints: 𝐴(𝑖,𝑗) 𝐷(𝑖) 𝑃(𝑦,𝑗) 𝑅(𝑗) 𝑇(𝑦,𝑗) 𝐹(𝑥,𝑦,𝑖,𝑗) 𝐻(𝑗) 𝐽(𝑦) 𝑉(𝑖,𝑗)

Proof of Theorem vdwlem6 Dummy variables 𝑚 𝑛 𝑧 𝑎 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . . 7 ⊢ (𝐺‘(𝐵 + (𝐸‘𝑖))) ∈ V
2		vdwlem6.j	. . . . . . 7 ⊢ 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸‘𝑖))))
3	1, 2	fnmpti 6022	. . . . . 6 ⊢ 𝐽 Fn (1...𝑀)
4		fvelrnb 6243	. . . . . 6 ⊢ (𝐽 Fn (1...𝑀) → ((𝐺‘𝐵) ∈ ran 𝐽 ↔ ∃𝑚 ∈ (1...𝑀)(𝐽‘𝑚) = (𝐺‘𝐵)))
5	3, 4	ax-mp 5	. . . . 5 ⊢ ((𝐺‘𝐵) ∈ ran 𝐽 ↔ ∃𝑚 ∈ (1...𝑀)(𝐽‘𝑚) = (𝐺‘𝐵))
6		vdwlem4.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Fin)
7	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝑅 ∈ Fin)
8		vdwlem7.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
9		eluz2nn 11726	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝐾 ∈ ℕ)
12		vdwlem3.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℕ)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝑊 ∈ ℕ)
14		vdwlem7.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝐺:(1...𝑊)⟶𝑅)
16		vdwlem6.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
17	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝐵 ∈ ℕ)
18		vdwlem7.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝑀 ∈ ℕ)
20		vdwlem6.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸:(1...𝑀)⟶ℕ)
21	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝐸:(1...𝑀)⟶ℕ)
22		vdwlem6.s	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
23	22	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
24		simprl 794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → 𝑚 ∈ (1...𝑀))
25		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → (𝐽‘𝑚) = (𝐺‘𝐵))
26		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → (𝐸‘𝑖) = (𝐸‘𝑚))
27	26	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝐵 + (𝐸‘𝑖)) = (𝐵 + (𝐸‘𝑚)))
28	27	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → (𝐺‘(𝐵 + (𝐸‘𝑖))) = (𝐺‘(𝐵 + (𝐸‘𝑚))))
29		fvex 6201	. . . . . . . . . 10 ⊢ (𝐺‘(𝐵 + (𝐸‘𝑚))) ∈ V
30	28, 2, 29	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑀) → (𝐽‘𝑚) = (𝐺‘(𝐵 + (𝐸‘𝑚))))
31	24, 30	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → (𝐽‘𝑚) = (𝐺‘(𝐵 + (𝐸‘𝑚))))
32	25, 31	eqtr3d 2658	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → (𝐺‘𝐵) = (𝐺‘(𝐵 + (𝐸‘𝑚))))
33	7, 11, 13, 15, 17, 19, 21, 23, 24, 32	vdwlem1 15685	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽‘𝑚) = (𝐺‘𝐵))) → (𝐾 + 1) MonoAP 𝐺)
34	33	rexlimdvaa 3032	. . . . 5 ⊢ (𝜑 → (∃𝑚 ∈ (1...𝑀)(𝐽‘𝑚) = (𝐺‘𝐵) → (𝐾 + 1) MonoAP 𝐺))
35	5, 34	syl5bi 232	. . . 4 ⊢ (𝜑 → ((𝐺‘𝐵) ∈ ran 𝐽 → (𝐾 + 1) MonoAP 𝐺))
36	35	imp 445	. . 3 ⊢ ((𝜑 ∧ (𝐺‘𝐵) ∈ ran 𝐽) → (𝐾 + 1) MonoAP 𝐺)
37	36	olcd 408	. 2 ⊢ ((𝜑 ∧ (𝐺‘𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
38		vdwlem3.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ ℕ)
39		vdwlem4.h	. . . . . . 7 ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
40		vdwlem4.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
41		vdwlem7.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ)
42		vdwlem7.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℕ)
43		vdwlem7.s	. . . . . . 7 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
44		vdwlem6.r	. . . . . . 7 ⊢ (𝜑 → (#‘ran 𝐽) = 𝑀)
45		vdwlem6.t	. . . . . . 7 ⊢ 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
46		vdwlem6.p	. . . . . . 7 ⊢ 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)))
47	38, 12, 6, 39, 40, 18, 14, 8, 41, 42, 43, 16, 20, 22, 2, 44, 45, 46	vdwlem5 15689	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℕ)
48	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝑇 ∈ ℕ)
49		0nn0 11307	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
50	49	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 = (𝑀 + 1)) → 0 ∈ ℕ₀)
51		nnuz 11723	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
52	18, 51	syl6eleq 2711	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
53	52	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝑀 ∈ (ℤ_≥‘1))
54		elfzp1 12391	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝑗 ∈ (1...(𝑀 + 1)) ↔ (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1))))
55	53, 54	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (𝑗 ∈ (1...(𝑀 + 1)) ↔ (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1))))
56	55	biimpa 501	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1)))
57	56	ord 392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 ∈ (1...𝑀) → 𝑗 = (𝑀 + 1)))
58	57	con1d 139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 = (𝑀 + 1) → 𝑗 ∈ (1...𝑀)))
59	58	imp 445	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → 𝑗 ∈ (1...𝑀))
60	20	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → 𝐸:(1...𝑀)⟶ℕ)
61	60	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸‘𝑗) ∈ ℕ)
62	61	nnnn0d 11351	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸‘𝑗) ∈ ℕ₀)
63	59, 62	syldan 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → (𝐸‘𝑗) ∈ ℕ₀)
64	50, 63	ifclda 4120	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) ∈ ℕ₀)
65	12, 42	nnmulcld 11068	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 · 𝐷) ∈ ℕ)
66	65	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑊 · 𝐷) ∈ ℕ)
67		nn0nnaddcl 11324	. . . . . . . 8 ⊢ ((if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) ∈ ℕ₀ ∧ (𝑊 · 𝐷) ∈ ℕ) → (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
68	64, 66, 67	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
69	68, 46	fmptd 6385	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝑃:(1...(𝑀 + 1))⟶ℕ)
70		nnex 11026	. . . . . . 7 ⊢ ℕ ∈ V
71		ovex 6678	. . . . . . 7 ⊢ (1...(𝑀 + 1)) ∈ V
72	70, 71	elmap 7886	. . . . . 6 ⊢ (𝑃 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1))) ↔ 𝑃:(1...(𝑀 + 1))⟶ℕ)
73	69, 72	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝑃 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1))))
74		elfzp1 12391	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝑖 ∈ (1...(𝑀 + 1)) ↔ (𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1))))
75	52, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) ↔ (𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1))))
76	16	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℕ)
77	76	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℂ)
78	77	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
79	20	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐸‘𝑖) ∈ ℕ)
80	79	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐸‘𝑖) ∈ ℂ)
81	80	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐸‘𝑖) ∈ ℂ)
82	78, 81	addcld 10059	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝐸‘𝑖)) ∈ ℂ)
83		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ₀)
84	41, 83	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐴 − 1) ∈ ℕ₀)
85		nn0nnaddcl 11324	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
86	84, 38, 85	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
87	12, 86	nnmulcld 11068	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
88	87	nncnd 11036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
89	88	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
90		elfznn0 12433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ₀)
91	90	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ₀)
92	91	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
93	92	adantlr 751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
94	93, 81	mulcld 10060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝐸‘𝑖)) ∈ ℂ)
95	65	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑊 · 𝐷) ∈ ℕ₀)
96	95	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℕ₀)
97	91, 96	nn0mulcld 11356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℕ₀)
98	97	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
99	98	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
100	82, 89, 94, 99	add4d 10264	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + ((𝑚 · (𝐸‘𝑖)) + (𝑚 · (𝑊 · 𝐷)))) = (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
101	65	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑊 · 𝐷) ∈ ℂ)
102	101	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℂ)
103	93, 81, 102	adddid 10064	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))) = ((𝑚 · (𝐸‘𝑖)) + (𝑚 · (𝑊 · 𝐷))))
104	103	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + ((𝑚 · (𝐸‘𝑖)) + (𝑚 · (𝑊 · 𝐷)))))
105	12	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑊 ∈ ℂ)
106	105	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℂ)
107	86	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
108	107	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 − 1) + 𝑉) ∈ ℂ)
109	42	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐷 ∈ ℂ)
110	109	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
111	92, 110	mulcld 10060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
112	106, 108, 111	adddid 10064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
113	41	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐴 ∈ ℂ)
114	113	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
115		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
116	114, 111, 115	addsubd 10413	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) − 1) = ((𝐴 − 1) + (𝑚 · 𝐷)))
117	116	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉))
118	84	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
119	118	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 − 1) ∈ ℂ)
120	38	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑉 ∈ ℂ)
121	120	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℂ)
122	119, 111, 121	add32d 10263	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
123	117, 122	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
124	123	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))))
125	92, 106, 110	mul12d 10245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) = (𝑊 · (𝑚 · 𝐷)))
126	125	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
127	112, 124, 126	3eqtr4d 2666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
128	127	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
129	128	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
130	100, 104, 129	3eqtr4d 2666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) = (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
131	38	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
132	12	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
133	43	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
134		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))
135		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑚 → (𝑛 · 𝐷) = (𝑚 · 𝐷))
136	135	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑚 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝑚 · 𝐷)))
137	136	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → ((𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)) ↔ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))))
138	137	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
139	134, 138	mpan2 707	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
140	10	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
141		vdwapval 15677	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
142	140, 41, 42, 141	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
143	142	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
144	139, 143	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
145	133, 144	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐹 “ {𝐺}))
146	38, 12, 6, 39, 40	vdwlem4 15688	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹:(1...𝑉)⟶(𝑅 ↑_𝑚 (1...𝑊)))
147		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹:(1...𝑉)⟶(𝑅 ↑_𝑚 (1...𝑊)) → 𝐹 Fn (1...𝑉))
148	146, 147	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 Fn (1...𝑉))
149		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 Fn (1...𝑉) → ((𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
150	148, 149	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
151	150	biimpa 501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐹 “ {𝐺})) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
152	145, 151	syldan 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
153	152	simpld 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
154	153	adantlr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
155	22	r19.21bi 2932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
156	155	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
157		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))
158		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑚 → (𝑛 · (𝐸‘𝑖)) = (𝑚 · (𝐸‘𝑖)))
159	158	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))))
160	159	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖))) ↔ ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))))
161	160	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖))))
162	157, 161	mpan2 707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖))))
163	10	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ)
164	163	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ₀)
165	76, 79	nnaddcld 11067	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸‘𝑖)) ∈ ℕ)
166		vdwapval 15677	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝐵 + (𝐸‘𝑖)) ∈ ℕ ∧ (𝐸‘𝑖) ∈ ℕ) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖)))))
167	164, 165, 79, 166	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖)))))
168	167	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) = ((𝐵 + (𝐸‘𝑖)) + (𝑛 · (𝐸‘𝑖)))) → ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)))
169	162, 168	sylan2 491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)))
170	156, 169	sseldd 3604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
171		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺:(1...𝑊)⟶𝑅 → 𝐺 Fn (1...𝑊))
172	14, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐺 Fn (1...𝑊))
173	172	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐺 Fn (1...𝑊))
174		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 Fn (1...𝑊) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ↔ (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
175	173, 174	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ↔ (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
176	175	biimpa 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))})) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
177	170, 176	syldan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
178	177	simpld 475	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊))
179	131, 132, 154, 178	vdwlem3 15687	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
180	130, 179	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))))
181		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) → (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
182	181	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
183		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
184		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
185	182, 183, 184	fvmpt 6282	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
186	178, 185	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
187	177	simprd 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))
188	152	simprd 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)
189		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑥 − 1) = ((𝐴 + (𝑚 · 𝐷)) − 1))
190	189	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → ((𝑥 − 1) + 𝑉) = (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))
191	190	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))
192	191	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
193	192	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
194	193	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
195		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...𝑊) ∈ V
196	195	mptex 6486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) ∈ V
197	194, 40, 196	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
198	153, 197	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
199	188, 198	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
200	199	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
201	200	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))))
202	187, 201	eqtr3d 2658	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝐵 + (𝐸‘𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖)))))
203	130	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))) = (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑚 · (𝐸‘𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
204	186, 202, 203	3eqtr4rd 2667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))
205	180, 204	jca 554	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
206		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉)))))
207		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → (𝐻‘𝑥) = (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))))
208	207	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → ((𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖))) ↔ (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
209	206, 208	anbi12d 747	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → ((𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖)))) ↔ ((((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘(((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
210	205, 209	syl5ibrcom 237	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
211	210	rexlimdva 3031	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
212	87	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
213	165, 212	nnaddcld 11067	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
214	65	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℕ)
215	79, 214	nnaddcld 11067	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐸‘𝑖) + (𝑊 · 𝐷)) ∈ ℕ)
216		vdwapval 15677	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ₀ ∧ ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ ∧ ((𝐸‘𝑖) + (𝑊 · 𝐷)) ∈ ℕ) → (𝑥 ∈ (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸‘𝑖) + (𝑊 · 𝐷))) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))))
217	164, 213, 215, 216	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸‘𝑖) + (𝑊 · 𝐷))) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸‘𝑖) + (𝑊 · 𝐷))))))
218		ffn 6045	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅 → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
219	39, 218	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
220	219	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
221		fniniseg 6338	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑥 ∈ (^◡𝐻 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
222	220, 221	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (^◡𝐻 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑥) = (𝐺‘(𝐵 + (𝐸‘𝑖))))))
223	211, 217, 222	3imtr4d 283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸‘𝑖) + (𝑊 · 𝐷))) → 𝑥 ∈ (^◡𝐻 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))})))
224	223	ssrdv 3609	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸‘𝑖) + (𝑊 · 𝐷))) ⊆ (^◡𝐻 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
225		ssun1 3776	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...𝑀) ⊆ ((1...𝑀) ∪ {(𝑀 + 1)})
226		fzsuc 12388	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
227	52, 226	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
228	225, 227	syl5sseqr 3654	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑀) ⊆ (1...(𝑀 + 1)))
229	228	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...(𝑀 + 1)))
230		eqeq1 2626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑖 → (𝑗 = (𝑀 + 1) ↔ 𝑖 = (𝑀 + 1)))
231		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑖 → (𝐸‘𝑗) = (𝐸‘𝑖))
232	230, 231	ifbieq2d 4111	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) = if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)))
233	232	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)) = (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)))
234		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)) ∈ V
235	233, 46, 234	fvmpt 6282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(𝑀 + 1)) → (𝑃‘𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)))
236	229, 235	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑃‘𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)))
237		elfzle2 12345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
238	18	nnred 11035	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ∈ ℝ)
239	238	ltp1d 10954	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
240		peano2re 10209	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
241	238, 240	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
242	238, 241	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
243	239, 242	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
244		breq1 4656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = (𝑀 + 1) → (𝑖 ≤ 𝑀 ↔ (𝑀 + 1) ≤ 𝑀))
245	244	notbid 308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = (𝑀 + 1) → (¬ 𝑖 ≤ 𝑀 ↔ ¬ (𝑀 + 1) ≤ 𝑀))
246	243, 245	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑖 = (𝑀 + 1) → ¬ 𝑖 ≤ 𝑀))
247	246	con2d 129	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑖 ≤ 𝑀 → ¬ 𝑖 = (𝑀 + 1)))
248	237, 247	syl5 34	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → ¬ 𝑖 = (𝑀 + 1)))
249	248	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ¬ 𝑖 = (𝑀 + 1))
250		iffalse 4095	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑖 = (𝑀 + 1) → if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) = (𝐸‘𝑖))
251	250	oveq1d 6665	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑖 = (𝑀 + 1) → (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)) = ((𝐸‘𝑖) + (𝑊 · 𝐷)))
252	249, 251	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (if(𝑖 = (𝑀 + 1), 0, (𝐸‘𝑖)) + (𝑊 · 𝐷)) = ((𝐸‘𝑖) + (𝑊 · 𝐷)))
253	236, 252	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑃‘𝑖) = ((𝐸‘𝑖) + (𝑊 · 𝐷)))
254	253	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃‘𝑖)) = (𝑇 + ((𝐸‘𝑖) + (𝑊 · 𝐷))))
255	47	nncnd 11036	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ ℂ)
256	255	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑇 ∈ ℂ)
257	101	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℂ)
258	256, 80, 257	add12d 10262	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑇 + ((𝐸‘𝑖) + (𝑊 · 𝐷))) = ((𝐸‘𝑖) + (𝑇 + (𝑊 · 𝐷))))
259	45	oveq1i 6660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 + (𝑊 · 𝐷)) = ((𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) + (𝑊 · 𝐷))
260	16	nncnd 11036	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℂ)
261	120, 109	subcld 10392	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑉 − 𝐷) ∈ ℂ)
262	113, 261	addcld 10059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐴 + (𝑉 − 𝐷)) ∈ ℂ)
263		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℂ
264		subcl 10280	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 + (𝑉 − 𝐷)) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + (𝑉 − 𝐷)) − 1) ∈ ℂ)
265	262, 263, 264	sylancl 694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐴 + (𝑉 − 𝐷)) − 1) ∈ ℂ)
266	105, 265	mulcld 10060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) ∈ ℂ)
267	260, 266, 101	addassd 10062	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + ((𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) + (𝑊 · 𝐷))))
268	105, 265, 109	adddid 10064	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑊 · (((𝐴 + (𝑉 − 𝐷)) − 1) + 𝐷)) = ((𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) + (𝑊 · 𝐷)))
269	113, 261, 109	addassd 10062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝐴 + (𝑉 − 𝐷)) + 𝐷) = (𝐴 + ((𝑉 − 𝐷) + 𝐷)))
270	120, 109	npcand 10396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑉 − 𝐷) + 𝐷) = 𝑉)
271	270	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐴 + ((𝑉 − 𝐷) + 𝐷)) = (𝐴 + 𝑉))
272	269, 271	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝐴 + (𝑉 − 𝐷)) + 𝐷) = (𝐴 + 𝑉))
273	272	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝐴 + (𝑉 − 𝐷)) + 𝐷) − 1) = ((𝐴 + 𝑉) − 1))
274		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ ℂ)
275	262, 109, 274	addsubd 10413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝐴 + (𝑉 − 𝐷)) + 𝐷) − 1) = (((𝐴 + (𝑉 − 𝐷)) − 1) + 𝐷))
276	113, 120, 274	addsubd 10413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴 + 𝑉) − 1) = ((𝐴 − 1) + 𝑉))
277	273, 275, 276	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝐴 + (𝑉 − 𝐷)) − 1) + 𝐷) = ((𝐴 − 1) + 𝑉))
278	277	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑊 · (((𝐴 + (𝑉 − 𝐷)) − 1) + 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
279	268, 278	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) + (𝑊 · 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
280	279	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 + ((𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)) + (𝑊 · 𝐷))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
281	267, 280	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
282	259, 281	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
283	282	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸‘𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸‘𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
284	283	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐸‘𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸‘𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
285	88	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
286	80, 77, 285	addassd 10062	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐸‘𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐸‘𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
287	80, 77	addcomd 10238	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐸‘𝑖) + 𝐵) = (𝐵 + (𝐸‘𝑖)))
288	287	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐸‘𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
289	284, 286, 288	3eqtr2d 2662	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐸‘𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
290	254, 258, 289	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃‘𝑖)) = ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
291	290, 253	oveq12d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) = (((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸‘𝑖) + (𝑊 · 𝐷))))
292		cnvimass 5485	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ⊆ dom 𝐺
293		fdm 6051	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(1...𝑊)⟶𝑅 → dom 𝐺 = (1...𝑊))
294	14, 293	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → dom 𝐺 = (1...𝑊))
295	292, 294	syl5sseq 3653	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ⊆ (1...𝑊))
296	295	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}) ⊆ (1...𝑊))
297		vdwapid1 15679	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℕ ∧ (𝐵 + (𝐸‘𝑖)) ∈ ℕ ∧ (𝐸‘𝑖) ∈ ℕ) → (𝐵 + (𝐸‘𝑖)) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)))
298	163, 165, 79, 297	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸‘𝑖)) ∈ ((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)))
299	155, 298	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸‘𝑖)) ∈ (^◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
300	296, 299	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸‘𝑖)) ∈ (1...𝑊))
301		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐵 + (𝐸‘𝑖)) → (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
302	301	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐵 + (𝐸‘𝑖)) → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
303		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
304		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻‘((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
305	302, 303, 304	fvmpt 6282	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 + (𝐸‘𝑖)) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸‘𝑖))) = (𝐻‘((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
306	300, 305	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸‘𝑖))) = (𝐻‘((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
307		vdwapid1 15679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
308	10, 41, 42, 307	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
309	43, 308	sseldd 3604	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ (^◡𝐹 “ {𝐺}))
310		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn (1...𝑉) → (𝐴 ∈ (^◡𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹‘𝐴) = 𝐺)))
311	148, 310	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐴 ∈ (^◡𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹‘𝐴) = 𝐺)))
312	309, 311	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∈ (1...𝑉) ∧ (𝐹‘𝐴) = 𝐺))
313	312	simprd 479	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐺)
314	312	simpld 475	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ (1...𝑉))
315		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1))
316	315	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) + 𝑉) = ((𝐴 − 1) + 𝑉))
317	316	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝐴 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
318	317	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝐴 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))
319	318	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝐴 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
320	319	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
321	195	mptex 6486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) ∈ V
322	320, 40, 321	fvmpt 6282	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ (1...𝑉) → (𝐹‘𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
323	314, 322	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹‘𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
324	313, 323	eqtr3d 2658	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
325	324	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺‘(𝐵 + (𝐸‘𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸‘𝑖))))
326	325	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐵 + (𝐸‘𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸‘𝑖))))
327	290	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃‘𝑖))) = (𝐻‘((𝐵 + (𝐸‘𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
328	306, 326, 327	3eqtr4rd 2667	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃‘𝑖))) = (𝐺‘(𝐵 + (𝐸‘𝑖))))
329	328	sneqd 4189	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → {(𝐻‘(𝑇 + (𝑃‘𝑖)))} = {(𝐺‘(𝐵 + (𝐸‘𝑖)))})
330	329	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) = (^◡𝐻 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))}))
331	224, 291, 330	3sstr4d 3648	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}))
332	331	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
333	260	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
334	88	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
335	333, 334, 98	addassd 10062	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
336	127	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
337	335, 336	eqtr4d 2659	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
338	38	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
339	12	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
340		eluzfz1 12348	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑀))
341	52, 340	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ (1...𝑀))
342		ne0i 3921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ (1...𝑀) → (1...𝑀) ≠ ∅)
343	341, 342	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1...𝑀) ≠ ∅)
344		elfzuz3 12339	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 + (𝐸‘𝑖)) ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘(𝐵 + (𝐸‘𝑖))))
345	300, 344	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ_≥‘(𝐵 + (𝐸‘𝑖))))
346	16	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐵 ∈ ℤ)
347		uzid 11702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ_≥‘𝐵))
348	346, 347	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘𝐵))
349	348	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐵 ∈ (ℤ_≥‘𝐵))
350	79	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐸‘𝑖) ∈ ℕ₀)
351		uzaddcl 11744	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ (ℤ_≥‘𝐵) ∧ (𝐸‘𝑖) ∈ ℕ₀) → (𝐵 + (𝐸‘𝑖)) ∈ (ℤ_≥‘𝐵))
352	349, 350, 351	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸‘𝑖)) ∈ (ℤ_≥‘𝐵))
353		uztrn 11704	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑊 ∈ (ℤ_≥‘(𝐵 + (𝐸‘𝑖))) ∧ (𝐵 + (𝐸‘𝑖)) ∈ (ℤ_≥‘𝐵)) → 𝑊 ∈ (ℤ_≥‘𝐵))
354	345, 352, 353	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ_≥‘𝐵))
355		eluzle 11700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑊 ∈ (ℤ_≥‘𝐵) → 𝐵 ≤ 𝑊)
356	354, 355	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐵 ≤ 𝑊)
357	356	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)𝐵 ≤ 𝑊)
358		r19.2z 4060	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑀) ≠ ∅ ∧ ∀𝑖 ∈ (1...𝑀)𝐵 ≤ 𝑊) → ∃𝑖 ∈ (1...𝑀)𝐵 ≤ 𝑊)
359	343, 357, 358	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∃𝑖 ∈ (1...𝑀)𝐵 ≤ 𝑊)
360		idd 24	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (1...𝑀) → (𝐵 ≤ 𝑊 → 𝐵 ≤ 𝑊))
361	360	rexlimiv 3027	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑖 ∈ (1...𝑀)𝐵 ≤ 𝑊 → 𝐵 ≤ 𝑊)
362	359, 361	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
363	12	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑊 ∈ ℤ)
364		fznn 12408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ ℤ → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵 ≤ 𝑊)))
365	363, 364	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵 ≤ 𝑊)))
366	16, 362, 365	mpbir2and 957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ (1...𝑊))
367	366	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ (1...𝑊))
368	338, 339, 153, 367	vdwlem3 15687	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
369	337, 368	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))))
370		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐵 → (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
371	370	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
372		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
373	371, 183, 372	fvmpt 6282	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
374	367, 373	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
375	199	fveq1d 6193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵))
376	337	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
377	374, 375, 376	3eqtr4rd 2667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺‘𝐵))
378	369, 377	jca 554	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺‘𝐵)))
379		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ↔ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉)))))
380		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝐻‘𝑧) = (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
381	380	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝐻‘𝑧) = (𝐺‘𝐵) ↔ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺‘𝐵)))
382	379, 381	anbi12d 747	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑧) = (𝐺‘𝐵)) ↔ (((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺‘𝐵))))
383	378, 382	syl5ibrcom 237	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑧) = (𝐺‘𝐵))))
384	383	rexlimdva 3031	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑧) = (𝐺‘𝐵))))
385	16, 87	nnaddcld 11067	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
386		vdwapval 15677	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ ∧ (𝑊 · 𝐷) ∈ ℕ) → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
387	140, 385, 65, 386	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
388		fniniseg 6338	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑧 ∈ (^◡𝐻 “ {(𝐺‘𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑧) = (𝐺‘𝐵))))
389	219, 388	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ (^◡𝐻 “ {(𝐺‘𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘𝑧) = (𝐺‘𝐵))))
390	384, 387, 389	3imtr4d 283	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) → 𝑧 ∈ (^◡𝐻 “ {(𝐺‘𝐵)})))
391	390	ssrdv 3609	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ⊆ (^◡𝐻 “ {(𝐺‘𝐵)}))
392	18	peano2nnd 11037	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
393	392, 51	syl6eleq 2711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ_≥‘1))
394		eluzfz2 12349	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘1) → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
395	393, 394	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
396		iftrue 4092	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑀 + 1) → if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) = 0)
397	396	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑀 + 1) → (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷)) = (0 + (𝑊 · 𝐷)))
398		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + (𝑊 · 𝐷)) ∈ V
399	397, 46, 398	fvmpt 6282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 + 1) ∈ (1...(𝑀 + 1)) → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
400	395, 399	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
401	101	addid2d 10237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 + (𝑊 · 𝐷)) = (𝑊 · 𝐷))
402	400, 401	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘(𝑀 + 1)) = (𝑊 · 𝐷))
403	402	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝑇 + (𝑊 · 𝐷)))
404	403, 282	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
405	404, 402	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)))
406		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
407	406	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
408		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
409	407, 303, 408	fvmpt 6282	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
410	366, 409	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
411	324	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺‘𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵))
412	404	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
413	410, 411, 412	3eqtr4rd 2667	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐺‘𝐵))
414	413	sneqd 4189	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))} = {(𝐺‘𝐵)})
415	414	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}) = (^◡𝐻 “ {(𝐺‘𝐵)}))
416	391, 405, 415	3sstr4d 3648	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
417		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑀 + 1) → (𝑃‘𝑖) = (𝑃‘(𝑀 + 1)))
418	417	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑀 + 1) → (𝑇 + (𝑃‘𝑖)) = (𝑇 + (𝑃‘(𝑀 + 1))))
419	418, 417	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) = ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))))
420	418	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑀 + 1) → (𝐻‘(𝑇 + (𝑃‘𝑖))) = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
421	420	sneqd 4189	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑀 + 1) → {(𝐻‘(𝑇 + (𝑃‘𝑖)))} = {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})
422	421	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑀 + 1) → (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) = (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
423	419, 422	sseq12d 3634	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑀 + 1) → (((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) ↔ ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})))
424	416, 423	syl5ibrcom 237	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
425	332, 424	jaod 395	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1)) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
426	75, 425	sylbid 230	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) → ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
427	426	ralrimiv 2965	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}))
428	427	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}))
429	227	rexeqdv 3145	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ ∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖)))))
430		rexun 3793	. . . . . . . . . . . . 13 ⊢ (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖)))))
431	328	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ 𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
432	431	rexbidva 3049	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖)))))
433		ovex 6678	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 + 1) ∈ V
434	420	eqeq2d 2632	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑀 + 1) → (𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))))
435	433, 434	rexsn 4223	. . . . . . . . . . . . . . 15 ⊢ (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
436	413	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) ↔ 𝑥 = (𝐺‘𝐵)))
437	435, 436	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ 𝑥 = (𝐺‘𝐵)))
438	432, 437	orbi12d 746	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖)))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))))
439	430, 438	syl5bb 272	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))))
440	429, 439	bitrd 268	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))))
441	440	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))))
442	441	abbidv 2741	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖)))} = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))})
443		eqid 2622	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))
444	443	rnmpt 5371	. . . . . . . . 9 ⊢ ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖)))) = {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃‘𝑖)))}
445	2	rnmpt 5371	. . . . . . . . . . 11 ⊢ ran 𝐽 = {𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖)))}
446		df-sn 4178	. . . . . . . . . . 11 ⊢ {(𝐺‘𝐵)} = {𝑥 ∣ 𝑥 = (𝐺‘𝐵)}
447	445, 446	uneq12i 3765	. . . . . . . . . 10 ⊢ (ran 𝐽 ∪ {(𝐺‘𝐵)}) = ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖)))} ∪ {𝑥 ∣ 𝑥 = (𝐺‘𝐵)})
448		unab 3894	. . . . . . . . . 10 ⊢ ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖)))} ∪ {𝑥 ∣ 𝑥 = (𝐺‘𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))}
449	447, 448	eqtri 2644	. . . . . . . . 9 ⊢ (ran 𝐽 ∪ {(𝐺‘𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸‘𝑖))) ∨ 𝑥 = (𝐺‘𝐵))}
450	442, 444, 449	3eqtr4g 2681	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖)))) = (ran 𝐽 ∪ {(𝐺‘𝐵)}))
451	450	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (#‘(ran 𝐽 ∪ {(𝐺‘𝐵)})))
452		fzfi 12771	. . . . . . . . . 10 ⊢ (1...𝑀) ∈ Fin
453		dffn4 6121	. . . . . . . . . . 11 ⊢ (𝐽 Fn (1...𝑀) ↔ 𝐽:(1...𝑀)–onto→ran 𝐽)
454	3, 453	mpbi 220	. . . . . . . . . 10 ⊢ 𝐽:(1...𝑀)–onto→ran 𝐽
455		fofi 8252	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ 𝐽:(1...𝑀)–onto→ran 𝐽) → ran 𝐽 ∈ Fin)
456	452, 454, 455	mp2an 708	. . . . . . . . 9 ⊢ ran 𝐽 ∈ Fin
457	456	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ran 𝐽 ∈ Fin)
458		fvex 6201	. . . . . . . . 9 ⊢ (𝐺‘𝐵) ∈ V
459		hashunsng 13181	. . . . . . . . 9 ⊢ ((𝐺‘𝐵) ∈ V → ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺‘𝐵)})) = ((#‘ran 𝐽) + 1)))
460	458, 459	ax-mp 5	. . . . . . . 8 ⊢ ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺‘𝐵)})) = ((#‘ran 𝐽) + 1))
461	457, 460	sylan 488	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺‘𝐵)})) = ((#‘ran 𝐽) + 1))
462	44	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘ran 𝐽) = 𝑀)
463	462	oveq1d 6665	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → ((#‘ran 𝐽) + 1) = (𝑀 + 1))
464	451, 461, 463	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (𝑀 + 1))
465	428, 464	jca 554	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (𝑀 + 1)))
466		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑎 = 𝑇 → (𝑎 + (𝑑‘𝑖)) = (𝑇 + (𝑑‘𝑖)))
467	466	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑎 = 𝑇 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
468	466	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑇 → (𝐻‘(𝑎 + (𝑑‘𝑖))) = (𝐻‘(𝑇 + (𝑑‘𝑖))))
469	468	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑎 = 𝑇 → {(𝐻‘(𝑎 + (𝑑‘𝑖)))} = {(𝐻‘(𝑇 + (𝑑‘𝑖)))})
470	469	imaeq2d 5466	. . . . . . . . 9 ⊢ (𝑎 = 𝑇 → (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}))
471	467, 470	sseq12d 3634	. . . . . . . 8 ⊢ (𝑎 = 𝑇 → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ↔ ((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))})))
472	471	ralbidv 2986	. . . . . . 7 ⊢ (𝑎 = 𝑇 → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ↔ ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))})))
473	468	mpteq2dv 4745	. . . . . . . . . 10 ⊢ (𝑎 = 𝑇 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖)))))
474	473	rneqd 5353	. . . . . . . . 9 ⊢ (𝑎 = 𝑇 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖)))))
475	474	fveq2d 6195	. . . . . . . 8 ⊢ (𝑎 = 𝑇 → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))))
476	475	eqeq1d 2624	. . . . . . 7 ⊢ (𝑎 = 𝑇 → ((#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (𝑀 + 1) ↔ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))) = (𝑀 + 1)))
477	472, 476	anbi12d 747	. . . . . 6 ⊢ (𝑎 = 𝑇 → ((∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))) = (𝑀 + 1))))
478		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑃 → (𝑑‘𝑖) = (𝑃‘𝑖))
479	478	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑑 = 𝑃 → (𝑇 + (𝑑‘𝑖)) = (𝑇 + (𝑃‘𝑖)))
480	479, 478	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑑 = 𝑃 → ((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)))
481	479	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑃 → (𝐻‘(𝑇 + (𝑑‘𝑖))) = (𝐻‘(𝑇 + (𝑃‘𝑖))))
482	481	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑑 = 𝑃 → {(𝐻‘(𝑇 + (𝑑‘𝑖)))} = {(𝐻‘(𝑇 + (𝑃‘𝑖)))})
483	482	imaeq2d 5466	. . . . . . . . 9 ⊢ (𝑑 = 𝑃 → (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}) = (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}))
484	480, 483	sseq12d 3634	. . . . . . . 8 ⊢ (𝑑 = 𝑃 → (((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}) ↔ ((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
485	484	ralbidv 2986	. . . . . . 7 ⊢ (𝑑 = 𝑃 → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}) ↔ ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))})))
486	481	mpteq2dv 4745	. . . . . . . . . 10 ⊢ (𝑑 = 𝑃 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖)))))
487	486	rneqd 5353	. . . . . . . . 9 ⊢ (𝑑 = 𝑃 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖)))))
488	487	fveq2d 6195	. . . . . . . 8 ⊢ (𝑑 = 𝑃 → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))) = (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))))
489	488	eqeq1d 2624	. . . . . . 7 ⊢ (𝑑 = 𝑃 → ((#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))) = (𝑀 + 1) ↔ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (𝑀 + 1)))
490	485, 489	anbi12d 747	. . . . . 6 ⊢ (𝑑 = 𝑃 → ((∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑‘𝑖))))) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (𝑀 + 1))))
491	477, 490	rspc2ev 3324	. . . . 5 ⊢ ((𝑇 ∈ ℕ ∧ 𝑃 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1))) ∧ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃‘𝑖))(AP‘𝐾)(𝑃‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑇 + (𝑃‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃‘𝑖))))) = (𝑀 + 1))) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (𝑀 + 1)))
492	48, 73, 465, 491	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (𝑀 + 1)))
493		ovex 6678	. . . . 5 ⊢ (1...(𝑊 · (2 · 𝑉))) ∈ V
494	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ)
495	494	nnnn0d 11351	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ₀)
496	39	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
497	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → 𝑀 ∈ ℕ)
498	497	peano2nnd 11037	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (𝑀 + 1) ∈ ℕ)
499		eqid 2622	. . . . 5 ⊢ (1...(𝑀 + 1)) = (1...(𝑀 + 1))
500	493, 495, 496, 498, 499	vdwpc 15684	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐻 “ {(𝐻‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑‘𝑖))))) = (𝑀 + 1))))
501	492, 500	mpbird 247	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → ⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻)
502	501	orcd 407	. 2 ⊢ ((𝜑 ∧ ¬ (𝐺‘𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
503	37, 502	pm2.61dan 832	1 ⊢ (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {csn 4177 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 #chash 13117 APcvdwa 15669 MonoAP cvdwm 15670 PolyAP cvdwp 15671

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-hash 13118 df-vdwap 15672 df-vdwmc 15673 df-vdwpc 15674

This theorem is referenced by: vdwlem7 15691

W3C validator