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Theorem ply1divex 23896

Description: Lemma for ply1divalg 23897: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)

**Hypotheses**
Ref	Expression
ply1divalg.p	⊢ 𝑃 = (Poly₁‘𝑅)
ply1divalg.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
ply1divalg.b	⊢ 𝐵 = (Base‘𝑃)
ply1divalg.m	⊢ − = (-_g‘𝑃)
ply1divalg.z	⊢ 0 = (0_g‘𝑃)
ply1divalg.t	⊢ ∙ = (._r‘𝑃)
ply1divalg.r1	⊢ (𝜑 → 𝑅 ∈ Ring)
ply1divalg.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
ply1divalg.g1	⊢ (𝜑 → 𝐺 ∈ 𝐵)
ply1divalg.g2	⊢ (𝜑 → 𝐺 ≠ 0 )
ply1divex.o	⊢ 1 = (1_r‘𝑅)
ply1divex.k	⊢ 𝐾 = (Base‘𝑅)
ply1divex.u	⊢ · = (._r‘𝑅)
ply1divex.i	⊢ (𝜑 → 𝐼 ∈ 𝐾)
ply1divex.g3	⊢ (𝜑 → (((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 )

**Assertion**
Ref	Expression
ply1divex	⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Distinct variable groups: 0 ,𝑞 𝐹,𝑞 𝐼,𝑞 𝑃,𝑞 𝑅,𝑞 − ,𝑞 𝐵,𝑞 ∙ ,𝑞 𝐷,𝑞 𝐺,𝑞 𝜑,𝑞 · ,𝑞 Allowed substitution hints: 1 (𝑞) 𝐾(𝑞)

Proof of Theorem ply1divex Dummy variables 𝑑 𝑓 𝑟 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 ⊢ (𝐹 = 0 → (𝐷‘𝐹) = (𝐷‘ 0 ))
2	1	breq1d 4663	. . . 4 ⊢ (𝐹 = 0 → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
3	2	rexbidv 3052	. . 3 ⊢ (𝐹 = 0 → (∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
4		nnssnn0 11295	. . . . 5 ⊢ ℕ ⊆ ℕ₀
5		ply1divalg.r1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Ring)
7		ply1divalg.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐵)
9		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝐹 ≠ 0 )
10		ply1divalg.d	. . . . . . . . . 10 ⊢ 𝐷 = ( deg₁ ‘𝑅)
11		ply1divalg.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly₁‘𝑅)
12		ply1divalg.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑃)
13		ply1divalg.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃)
14	10, 11, 12, 13	deg1nn0cl 23848	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
15	6, 8, 9, 14	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
16	15	nn0red 11352	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℝ)
17		ply1divalg.g1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
18		ply1divalg.g2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ≠ 0 )
19	10, 11, 12, 13	deg1nn0cl 23848	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
20	5, 17, 18, 19	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
21	20	nn0red 11352	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ)
22	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐺) ∈ ℝ)
23	16, 22	resubcld 10458	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ((𝐷‘𝐹) − (𝐷‘𝐺)) ∈ ℝ)
24		arch 11289	. . . . . 6 ⊢ (((𝐷‘𝐹) − (𝐷‘𝐺)) ∈ ℝ → ∃𝑑 ∈ ℕ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑)
25	23, 24	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ∃𝑑 ∈ ℕ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑)
26		ssrexv 3667	. . . . 5 ⊢ (ℕ ⊆ ℕ₀ → (∃𝑑 ∈ ℕ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 → ∃𝑑 ∈ ℕ₀ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑))
27	4, 25, 26	mpsyl 68	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ∃𝑑 ∈ ℕ₀ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑)
28	16	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐹) ∈ ℝ)
29	21	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℝ)
30		nn0re 11301	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℝ)
31	30	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℝ)
32	28, 29, 31	ltsubadd2d 10625	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 ↔ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
33	32	biimpd 219	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 → (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
34	33	reximdva 3017	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (∃𝑑 ∈ ℕ₀ ((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 → ∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
35	27, 34	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑))
36		0nn0 11307	. . . 4 ⊢ 0 ∈ ℕ₀
37	10, 11, 12	deg1z 23847	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞)
38	5, 37	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷‘ 0 ) = -∞)
39		0re 10040	. . . . . . 7 ⊢ 0 ∈ ℝ
40		readdcl 10019	. . . . . . 7 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐷‘𝐺) + 0) ∈ ℝ)
41	21, 39, 40	sylancl 694	. . . . . 6 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) ∈ ℝ)
42		mnflt 11957	. . . . . 6 ⊢ (((𝐷‘𝐺) + 0) ∈ ℝ → -∞ < ((𝐷‘𝐺) + 0))
43	41, 42	syl 17	. . . . 5 ⊢ (𝜑 → -∞ < ((𝐷‘𝐺) + 0))
44	38, 43	eqbrtrd 4675	. . . 4 ⊢ (𝜑 → (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0))
45		oveq2 6658	. . . . . 6 ⊢ (𝑑 = 0 → ((𝐷‘𝐺) + 𝑑) = ((𝐷‘𝐺) + 0))
46	45	breq2d 4665	. . . . 5 ⊢ (𝑑 = 0 → ((𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)))
47	46	rspcev 3309	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)) → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
48	36, 44, 47	sylancr 695	. . 3 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
49	3, 35, 48	pm2.61ne 2879	. 2 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑))
50	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝐹 ∈ 𝐵)
51		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑎 = 0 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 0))
52	51	breq2d 4665	. . . . . . . . 9 ⊢ (𝑎 = 0 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)))
53	52	imbi1d 331	. . . . . . . 8 ⊢ (𝑎 = 0 → (((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
54	53	ralbidv 2986	. . . . . . 7 ⊢ (𝑎 = 0 → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
55	54	imbi2d 330	. . . . . 6 ⊢ (𝑎 = 0 → ((𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) ↔ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
56		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 𝑑))
57	56	breq2d 4665	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑)))
58	57	imbi1d 331	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
59	58	ralbidv 2986	. . . . . . 7 ⊢ (𝑎 = 𝑑 → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
60	59	imbi2d 330	. . . . . 6 ⊢ (𝑎 = 𝑑 → ((𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) ↔ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
61		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + (𝑑 + 1)))
62	61	breq2d 4665	. . . . . . . . 9 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
63	62	imbi1d 331	. . . . . . . 8 ⊢ (𝑎 = (𝑑 + 1) → (((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
64	63	ralbidv 2986	. . . . . . 7 ⊢ (𝑎 = (𝑑 + 1) → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
65	64	imbi2d 330	. . . . . 6 ⊢ (𝑎 = (𝑑 + 1) → ((𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) ↔ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
66	11	ply1ring 19618	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
67	5, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ Ring)
68	13, 12	ring0cl 18569	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 0 ∈ 𝐵)
69	67, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
70	69	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → 0 ∈ 𝐵)
71		ply1divalg.t	. . . . . . . . . . . . . . . . 17 ⊢ ∙ = (._r‘𝑃)
72	13, 71, 12	ringrz 18588	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∙ 0 ) = 0 )
73	67, 17, 72	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ∙ 0 ) = 0 )
74	73	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
75	74	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
76		ringgrp 18552	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
77	67, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Grp)
78		ply1divalg.m	. . . . . . . . . . . . . . 15 ⊢ − = (-_g‘𝑃)
79	13, 12, 78	grpsubid1 17500	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ Grp ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
80	77, 79	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
81	75, 80	eqtr2d 2657	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 = (𝑓 − (𝐺 ∙ 0 )))
82	81	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐷‘𝑓) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
83	20	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℂ)
84	83	addid1d 10236	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
85	84	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
86	82, 85	breq12d 4666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
87	86	biimpa 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺))
88		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑞 = 0 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 0 ))
89	88	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑞 = 0 → (𝑓 − (𝐺 ∙ 𝑞)) = (𝑓 − (𝐺 ∙ 0 )))
90	89	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑞 = 0 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
91	90	breq1d 4663	. . . . . . . . . 10 ⊢ (𝑞 = 0 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
92	91	rspcev 3309	. . . . . . . . 9 ⊢ (( 0 ∈ 𝐵 ∧ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
93	70, 87, 92	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
94	93	ex 450	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
95	94	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
96		nn0addcl 11328	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷‘𝐺) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
97	20, 96	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
98	97	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
99	5	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑅 ∈ Ring)
100		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑔 ∈ 𝐵)
101	10, 11, 13	deg1cl 23843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 ∈ 𝐵 → (𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}))
102	101	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}))
103	20	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℕ₀)
104		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
105	104	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝑑 + 1) ∈ ℕ₀)
106	103, 105	nn0addcld 11355	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℕ₀)
107	106	nn0zd 11480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℤ)
108		degltlem1 23832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}) ∧ ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℤ) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
109	102, 107, 108	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
110	109	biimpd 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
111	110	impr 649	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1))
112	20	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℕ₀)
113	112	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℂ)
114		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℂ)
115	114	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℂ)
116		peano2cn 10208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
117	115, 116	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑑 + 1) ∈ ℂ)
118		1cnd 10056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 1 ∈ ℂ)
119	113, 117, 118	addsubassd 10412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + ((𝑑 + 1) − 1)))
120		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
121		pncan 10287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑑 + 1) − 1) = 𝑑)
122	115, 120, 121	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑑 + 1) − 1) = 𝑑)
123	122	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + ((𝑑 + 1) − 1)) = ((𝐷‘𝐺) + 𝑑))
124	119, 123	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
125	124	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
126	111, 125	breqtrd 4679	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ ((𝐷‘𝐺) + 𝑑))
127	67	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Ring)
128	17	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐺 ∈ 𝐵)
129	5	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑅 ∈ Ring)
130		ply1divex.i	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ 𝐾)
131	130	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝐾)
132		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑔) = (coe₁‘𝑔)
133		ply1divex.k	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐾 = (Base‘𝑅)
134	132, 13, 11, 133	coe1f 19581	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 ∈ 𝐵 → (coe₁‘𝑔):ℕ₀⟶𝐾)
135	134	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝑔):ℕ₀⟶𝐾)
136		simplr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℕ₀)
137	103, 136	nn0addcld 11355	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
138	135, 137	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾)
139		ply1divex.u	. . . . . . . . . . . . . . . . . . . . 21 ⊢ · = (._r‘𝑅)
140	133, 139	ringcl 18561	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐾 ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
141	129, 131, 138, 140	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
142		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
143		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
144		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
145		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
146	133, 11, 142, 143, 144, 145, 13	ply1tmcl 19642	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾 ∧ 𝑑 ∈ ℕ₀) → ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)
147	129, 141, 136, 146	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)
148	13, 71	ringcl 18561	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
149	127, 128, 147, 148	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
150	149	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
151	103	nn0red 11352	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℝ)
152	151	leidd 10594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ≤ (𝐷‘𝐺))
153	10, 133, 11, 142, 143, 144, 145	deg1tmle 23877	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ≤ 𝑑)
154	129, 141, 136, 153	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ≤ 𝑑)
155	11, 10, 129, 13, 71, 128, 147, 103, 136, 152, 154	deg1mulle2 23869	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ≤ ((𝐷‘𝐺) + 𝑑))
156	155	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ≤ ((𝐷‘𝐺) + 𝑑))
157		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) = (coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))
158		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
159	158, 133, 11, 142, 143, 144, 145, 13, 71, 139, 128, 129, 141, 136, 103	coe1tmmul2fv 19648	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))‘(𝑑 + (𝐷‘𝐺))) = (((coe₁‘𝐺)‘(𝐷‘𝐺)) · (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))))
160	103	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℂ)
161	114	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℂ)
162	160, 161	addcomd 10238	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) = (𝑑 + (𝐷‘𝐺)))
163	162	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))‘((𝐷‘𝐺) + 𝑑)) = ((coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))‘(𝑑 + (𝐷‘𝐺))))
164		ply1divex.g3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 )
165	164	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
166	165	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
167		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
168	167, 13, 11, 133	coe1f 19581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ 𝐵 → (coe₁‘𝐺):ℕ₀⟶𝐾)
169	17, 168	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (coe₁‘𝐺):ℕ₀⟶𝐾)
170	169	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝐺):ℕ₀⟶𝐾)
171	170, 103	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐾)
172	133, 139	ringass 18564	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ (((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐾 ∧ 𝐼 ∈ 𝐾 ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾)) → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = (((coe₁‘𝐺)‘(𝐷‘𝐺)) · (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))))
173	129, 171, 131, 138, 172	syl13anc 1328	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = (((coe₁‘𝐺)‘(𝐷‘𝐺)) · (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))))
174		ply1divex.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 = (1_r‘𝑅)
175	133, 139, 174	ringlidm 18571	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾) → ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))
176	129, 138, 175	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))
177	166, 173, 176	3eqtr3rd 2665	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) = (((coe₁‘𝐺)‘(𝐷‘𝐺)) · (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))))
178	159, 163, 177	3eqtr4rd 2667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) = ((coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))‘((𝐷‘𝐺) + 𝑑)))
179	178	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) = ((coe₁‘(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))‘((𝐷‘𝐺) + 𝑑)))
180	10, 11, 13, 78, 98, 99, 100, 126, 150, 156, 132, 157, 179	deg1sublt 23870	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑))
181	180	adantlrr 757	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑))
182	77	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Grp)
183		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
184	13, 78	grpsubcl 17495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Grp ∧ 𝑔 ∈ 𝐵 ∧ (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵) → (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵)
185	182, 183, 149, 184	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵)
186	185	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵)
187	186	adantlrr 757	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵)
188		simplrr 801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
189		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → (𝐷‘𝑓) = (𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
190	189	breq1d 4663	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑)))
191		oveq1 6657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → (𝑓 − (𝐺 ∙ 𝑞)) = ((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞)))
192	191	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))))
193	192	breq1d 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
194	193	rexbidv 3052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → (∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
195	190, 194	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) → (((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ((𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
196	195	rspcva 3307	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵 ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) → ((𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
197	187, 188, 196	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ((𝐷‘(𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
198	181, 197	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
199	67	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Ring)
200		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵)
201	147	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)
202		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
203	13, 202	ringacl 18578	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵) → (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
204	199, 200, 201, 203	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
205	77	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Grp)
206		simplr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑔 ∈ 𝐵)
207	149	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
208	17	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵)
209	13, 71	ringcl 18561	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
210	199, 208, 200, 209	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
211	13, 202, 78	grpsubsub4 17508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ Grp ∧ (𝑔 ∈ 𝐵 ∧ (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵 ∧ (𝐺 ∙ 𝑞) ∈ 𝐵)) → ((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞)) = (𝑔 − ((𝐺 ∙ 𝑞)(+_g‘𝑃)(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
212	205, 206, 207, 210, 211	syl13anc 1328	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞)) = (𝑔 − ((𝐺 ∙ 𝑞)(+_g‘𝑃)(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
213	13, 202, 71	ringdi 18566	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ Ring ∧ (𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵 ∧ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)) → (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) = ((𝐺 ∙ 𝑞)(+_g‘𝑃)(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
214	199, 208, 200, 201, 213	syl13anc 1328	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) = ((𝐺 ∙ 𝑞)(+_g‘𝑃)(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
215	214	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))) = (𝑔 − ((𝐺 ∙ 𝑞)(+_g‘𝑃)(𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
216	212, 215	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞)) = (𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
217	216	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) = (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))))
218	217	breq1d 4663	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))) < (𝐷‘𝐺)))
219	218	biimpd 219	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) → (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))) < (𝐷‘𝐺)))
220		oveq2 6658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) → (𝐺 ∙ 𝑟) = (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
221	220	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) → (𝑔 − (𝐺 ∙ 𝑟)) = (𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))))
222	221	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) → (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))))
223	222	breq1d 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) → ((𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))) < (𝐷‘𝐺)))
224	223	rspcev 3309	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵 ∧ (𝐷‘(𝑔 − (𝐺 ∙ (𝑞(+_g‘𝑃)((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))) < (𝐷‘𝐺)) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))
225	204, 219, 224	syl6an 568	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
226	225	rexlimdva 3031	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
227	226	adantrr 753	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
228	227	adantlrr 757	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (∃𝑞 ∈ 𝐵 (𝐷‘((𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
229	198, 228	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))
230	229	expr 643	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
231	230	ralrimiva 2966	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
232		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝐷‘𝑔) = (𝐷‘𝑓))
233	232	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
234		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (𝑔 − (𝐺 ∙ 𝑟)) = (𝑓 − (𝐺 ∙ 𝑟)))
235	234	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))))
236	235	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
237	236	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
238		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑞 → (𝐺 ∙ 𝑟) = (𝐺 ∙ 𝑞))
239	238	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑞 → (𝑓 − (𝐺 ∙ 𝑟)) = (𝑓 − (𝐺 ∙ 𝑞)))
240	239	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑞 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))))
241	240	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
242	241	cbvrexv 3172	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
243	237, 242	syl6bb 276	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
244	233, 243	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) ↔ ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
245	244	cbvralv 3171	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
246	231, 245	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
247	246	exp32 631	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ ℕ₀ → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
248	247	com12 32	. . . . . . 7 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
249	248	a2d 29	. . . . . 6 ⊢ (𝑑 ∈ ℕ₀ → ((𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) → (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))))
250	55, 60, 65, 60, 95, 249	nn0ind 11472	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
251	250	impcom 446	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
252		fveq2 6191	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
253	252	breq1d 4663	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
254		oveq1 6657	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 − (𝐺 ∙ 𝑞)) = (𝐹 − (𝐺 ∙ 𝑞)))
255	254	fveq2d 6195	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))))
256	255	breq1d 4663	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
257	256	rexbidv 3052	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
258	253, 257	imbi12d 334	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) ↔ ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
259	258	rspcva 3307	. . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
260	50, 251, 259	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
261	260	rexlimdva 3031	. 2 ⊢ (𝜑 → (∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
262	49, 261	mpd 15	1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∪ cun 3572 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 -∞cmnf 10072 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 Basecbs 15857 +_gcplusg 15941 ._rcmulr 15942 ·_𝑠 cvsca 15945 0_gc0g 16100 Grpcgrp 17422 -_gcsg 17424 ._gcmg 17540 mulGrpcmgp 18489 1_rcur 18501 Ringcrg 18547 var₁cv1 19546 Poly₁cpl1 19547 coe₁cco1 19548 deg₁ cdg1 23814

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-inf2 8538 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 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

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-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 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-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-subrg 18778 df-lmod 18865 df-lss 18933 df-rlreg 19283 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-cnfld 19747 df-mdeg 23815 df-deg1 23816

This theorem is referenced by: ply1divalg 23897

W3C validator