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Theorem vdwlem2 15686

Description: Lemma for vdw 15698. (Contributed by Mario Carneiro, 12-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem2.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem2.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)
vdwlem2.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem2.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
vdwlem2.f	⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
vdwlem2.m	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
vdwlem2.g	⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))

**Assertion**
Ref	Expression
vdwlem2	⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐾 𝑥,𝑀 𝜑,𝑥 𝑥,𝐺 𝑥,𝑁 𝑥,𝑅 𝑥,𝑊

Proof of Theorem vdwlem2 Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ)
2		vdwlem2.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		nnaddcl 11042	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
4	1, 2, 3	syl2anr 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5		simpllr 799	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
6	5	nncnd 11036	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
7	2	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
8	7	nncnd 11036	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9		elfznn0 12433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ₀)
10	9	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ₀)
11	10	nn0cnd 11353	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12		simplrl 800	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
13	12	nncnd 11036	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
14	11, 13	mulcld 10060	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
15	6, 8, 14	add32d 10263	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
16		simplrr 801	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))
17		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
18		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
19	18	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
20	19	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → ((𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)) ↔ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))))
21	20	rspcev 3309	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
22	17, 21	mpan2 707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
23	22	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
24		vdwlem2.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
25	24	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ₀)
26	25	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ₀)
27		vdwapval 15677	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
28	26, 5, 12, 27	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
29	23, 28	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
30	16, 29	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}))
31		elfznn 12370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
32		nnaddcl 11042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
33	31, 2, 32	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
34		nnuz 11723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ_≥‘1)
35	33, 34	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ_≥‘1))
36		vdwlem2.m	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
37	36	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
38		elfzuz3 12339	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘𝑥))
39	2	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℤ)
40		eluzadd 11716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ (ℤ_≥‘𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
41	38, 39, 40	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
42		uztrn 11704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
43	37, 41, 42	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
44		elfzuzb 12336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁))))
45	35, 43, 44	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
46		vdwlem2.f	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
47	46	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
48	45, 47	syldan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
49		vdwlem2.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
50	48, 49	fmptd 6385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
51		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺:(1...𝑊)⟶𝑅 → 𝐺 Fn (1...𝑊))
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 Fn (1...𝑊))
53	52	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
54		fniniseg 6338	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn (1...𝑊) → ((𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
55	53, 54	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
56	30, 55	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐))
57	56	simpld 475	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
58	45	ralrimiva 2966	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
59	58	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
60		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
61	60	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
62	61	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
63	57, 59, 62	sylc 65	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
64	15, 63	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
65	15	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
66	60	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
67		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
68	66, 49, 67	fvmpt 6282	. . . . . . . . . . . . . 14 ⊢ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
69	57, 68	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
70	56	simprd 479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
71	65, 69, 70	3eqtr2d 2662	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
72	64, 71	jca 554	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
73		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
74		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝐹‘𝑥) = (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))))
75	74	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹‘𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
76	73, 75	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
77	72, 76	syl5ibrcom 237	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
78	77	rexlimdva 3031	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
79	4	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
80		simprl 794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
81		vdwapval 15677	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
82	25, 79, 80, 81	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
83		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐹:(1...𝑀)⟶𝑅 → 𝐹 Fn (1...𝑀))
84	46, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (1...𝑀))
85	84	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
86		fniniseg 6338	. . . . . . . . . 10 ⊢ (𝐹 Fn (1...𝑀) → (𝑥 ∈ (^◡𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
87	85, 86	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ (^◡𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
88	78, 82, 87	3imtr4d 283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (^◡𝐹 “ {𝑐})))
89	88	ssrdv 3609	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
90	89	expr 643	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
91	90	reximdva 3017	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
92		oveq1 6657	. . . . . . . 8 ⊢ (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
93	92	sseq1d 3632	. . . . . . 7 ⊢ (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
94	93	rexbidv 3052	. . . . . 6 ⊢ (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
95	94	rspcev 3309	. . . . 5 ⊢ (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
96	4, 91, 95	syl6an 568	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
97	96	rexlimdva 3031	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
98	97	eximdv 1846	. 2 ⊢ (𝜑 → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
99		ovex 6678	. . 3 ⊢ (1...𝑊) ∈ V
100	99, 24, 50	vdwmc 15682	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐})))
101		ovex 6678	. . 3 ⊢ (1...𝑀) ∈ V
102	101, 24, 46	vdwmc 15682	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
103	98, 100, 102	3imtr4d 283	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 APcvdwa 15669 MonoAP cvdwm 15670

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-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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-vdwap 15672 df-vdwmc 15673

This theorem is referenced by: vdwlem9 15693

W3C validator