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Theorem ply1divex 26065

Description: Lemma for ply1divalg 26066: 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 6891	. . . . 5 ⊢ (𝐹 = 0 → (𝐷‘𝐹) = (𝐷‘ 0 ))
2	1	breq1d 5152	. . . 4 ⊢ (𝐹 = 0 → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
3	2	rexbidv 3173	. . 3 ⊢ (𝐹 = 0 → (∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
4		nnssnn0 12499	. . . . 5 ⊢ ℕ ⊆ ℕ₀
5		ply1divalg.r1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Ring)
7		ply1divalg.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐵)
9		simpr 484	. . . . . . . . 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 26017	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
15	6, 8, 9, 14	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
16	15	nn0red 12557	. . . . . . 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 26017	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
20	5, 17, 18, 19	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
21	20	nn0red 12557	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐺) ∈ ℝ)
23	16, 22	resubcld 11666	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ((𝐷‘𝐹) − (𝐷‘𝐺)) ∈ ℝ)
24		arch 12493	. . . . . 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 4047	. . . . 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 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐹) ∈ ℝ)
29	21	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℝ)
30		nn0re 12505	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℝ)
31	30	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℝ)
32	28, 29, 31	ltsubadd2d 11836	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 ↔ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
33	32	biimpd 228	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐹) − (𝐷‘𝐺)) < 𝑑 → (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
34	33	reximdva 3163	. . . 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 12511	. . . 4 ⊢ 0 ∈ ℕ₀
37	10, 11, 12	deg1z 26016	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞)
38	5, 37	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷‘ 0 ) = -∞)
39		0re 11240	. . . . . . 7 ⊢ 0 ∈ ℝ
40		readdcl 11215	. . . . . . 7 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐷‘𝐺) + 0) ∈ ℝ)
41	21, 39, 40	sylancl 585	. . . . . 6 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) ∈ ℝ)
42	41	mnfltd 13130	. . . . 5 ⊢ (𝜑 → -∞ < ((𝐷‘𝐺) + 0))
43	38, 42	eqbrtrd 5164	. . . 4 ⊢ (𝜑 → (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0))
44		oveq2 7422	. . . . . 6 ⊢ (𝑑 = 0 → ((𝐷‘𝐺) + 𝑑) = ((𝐷‘𝐺) + 0))
45	44	breq2d 5154	. . . . 5 ⊢ (𝑑 = 0 → ((𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)))
46	45	rspcev 3607	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)) → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
47	36, 43, 46	sylancr 586	. . 3 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
48	3, 35, 47	pm2.61ne 3022	. 2 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑))
49		fveq2 6891	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
50	49	breq1d 5152	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
51		fvoveq1 7437	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))))
52	51	breq1d 5152	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
53	52	rexbidv 3173	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
54	50, 53	imbi12d 344	. . . 4 ⊢ (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))))
55		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑎 = 0 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 0))
56	55	breq2d 5154	. . . . . . . . 9 ⊢ (𝑎 = 0 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)))
57	56	imbi1d 341	. . . . . . . 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))))
58	57	ralbidv 3172	. . . . . . 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))))
59	58	imbi2d 340	. . . . . 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)))))
60		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 𝑑))
61	60	breq2d 5154	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑)))
62	61	imbi1d 341	. . . . . . . 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))))
63	62	ralbidv 3172	. . . . . . 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))))
64	63	imbi2d 340	. . . . . 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)))))
65		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + (𝑑 + 1)))
66	65	breq2d 5154	. . . . . . . . 9 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
67	66	imbi1d 341	. . . . . . . 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))))
68	67	ralbidv 3172	. . . . . . 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))))
69	68	imbi2d 340	. . . . . 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)))))
70	11	ply1ring 22159	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
71	5, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ Ring)
72	13, 12	ring0cl 20196	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 0 ∈ 𝐵)
73	71, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
74	73	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → 0 ∈ 𝐵)
75		ply1divalg.t	. . . . . . . . . . . . . . . . 17 ⊢ ∙ = (._r‘𝑃)
76	13, 75, 12	ringrz 20223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∙ 0 ) = 0 )
77	71, 17, 76	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ∙ 0 ) = 0 )
78	77	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
79	78	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
80		ringgrp 20171	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
81	71, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Grp)
82		ply1divalg.m	. . . . . . . . . . . . . . 15 ⊢ − = (-_g‘𝑃)
83	13, 12, 82	grpsubid1 18974	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ Grp ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
84	81, 83	sylan 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
85	79, 84	eqtr2d 2768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 = (𝑓 − (𝐺 ∙ 0 )))
86	85	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐷‘𝑓) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
87	20	nn0cnd 12558	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℂ)
88	87	addridd 11438	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
89	88	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
90	86, 89	breq12d 5155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
91	90	biimpa 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺))
92		oveq2 7422	. . . . . . . . . . . . 13 ⊢ (𝑞 = 0 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 0 ))
93	92	oveq2d 7430	. . . . . . . . . . . 12 ⊢ (𝑞 = 0 → (𝑓 − (𝐺 ∙ 𝑞)) = (𝑓 − (𝐺 ∙ 0 )))
94	93	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑞 = 0 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
95	94	breq1d 5152	. . . . . . . . . 10 ⊢ (𝑞 = 0 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
96	95	rspcev 3607	. . . . . . . . 9 ⊢ (( 0 ∈ 𝐵 ∧ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
97	74, 91, 96	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
98	97	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
99	98	ralrimiva 3141	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
100		nn0addcl 12531	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷‘𝐺) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
101	20, 100	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
102	101	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
103	5	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑅 ∈ Ring)
104		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑔 ∈ 𝐵)
105	10, 11, 13	deg1cl 26012	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 ∈ 𝐵 → (𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}))
106	20	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℕ₀)
107		peano2nn0 12536	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
108	107	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝑑 + 1) ∈ ℕ₀)
109	106, 108	nn0addcld 12560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℕ₀)
110	109	nn0zd 12608	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℤ)
111		degltlem1 26001	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}) ∧ ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℤ) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
112	105, 110, 111	syl2an2 685	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
113	112	biimpd 228	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) → (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1)))
114	113	impr 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1))
115	20	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℕ₀)
116	115	nn0cnd 12558	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℂ)
117		nn0cn 12506	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℂ)
118	117	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℂ)
119		peano2cn 11410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
120	118, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑑 + 1) ∈ ℂ)
121		1cnd 11233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 1 ∈ ℂ)
122	116, 120, 121	addsubassd 11615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + ((𝑑 + 1) − 1)))
123		ax-1cn 11190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
124		pncan 11490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑑 + 1) − 1) = 𝑑)
125	118, 123, 124	sylancl 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑑 + 1) − 1) = 𝑑)
126	125	oveq2d 7430	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + ((𝑑 + 1) − 1)) = ((𝐷‘𝐺) + 𝑑))
127	122, 126	eqtrd 2767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
128	127	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
129	114, 128	breqtrd 5168	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ ((𝐷‘𝐺) + 𝑑))
130	71	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Ring)
131	17	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐺 ∈ 𝐵)
132	5	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑅 ∈ Ring)
133		ply1divex.i	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ 𝐾)
134	133	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝐾)
135		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑔) = (coe₁‘𝑔)
136		ply1divex.k	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐾 = (Base‘𝑅)
137	135, 13, 11, 136	coe1f 22123	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 ∈ 𝐵 → (coe₁‘𝑔):ℕ₀⟶𝐾)
138	137	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝑔):ℕ₀⟶𝐾)
139		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℕ₀)
140	106, 139	nn0addcld 12560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
141	138, 140	ffvelcdmd 7089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾)
142		ply1divex.u	. . . . . . . . . . . . . . . . . . . . 21 ⊢ · = (._r‘𝑅)
143	136, 142	ringcl 20183	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐾 ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
144	132, 134, 141, 143	syl3anc 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
145		eqid 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
146		eqid 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
147		eqid 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
148		eqid 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
149	136, 11, 145, 146, 147, 148, 13	ply1tmcl 22184	. . . . . . . . . . . . . . . . . . 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)
150	132, 144, 139, 149	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)
151	13, 75	ringcl 20183	. . . . . . . . . . . . . . . . . 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)
152	130, 131, 150, 151	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
153	152	adantrr 716	. . . . . . . . . . . . . . . 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)
154	106	nn0red 12557	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℝ)
155	154	leidd 11804	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ≤ (𝐷‘𝐺))
156	10, 136, 11, 145, 146, 147, 148	deg1tmle 26046	. . . . . . . . . . . . . . . . . . 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)
157	132, 144, 139, 156	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ≤ 𝑑)
158	11, 10, 132, 13, 75, 131, 150, 106, 139, 155, 157	deg1mulle2 26038	. . . . . . . . . . . . . . . . 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))
159	158	adantrr 716	. . . . . . . . . . . . . . . 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))
160		eqid 2727	. . . . . . . . . . . . . . . 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)))))
161		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
162	161, 136, 11, 145, 146, 147, 148, 13, 75, 142, 131, 132, 144, 139, 106	coe1tmmul2fv 22190	. . . . . . . . . . . . . . . . . 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)))))
163	106	nn0cnd 12558	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℂ)
164	117	ad2antlr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℂ)
165	163, 164	addcomd 11440	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) = (𝑑 + (𝐷‘𝐺)))
166	165	fveq2d 6895	. . . . . . . . . . . . . . . . . 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))))
167		ply1divex.g3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 )
168	167	oveq1d 7429	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
169	168	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
170		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
171	170, 13, 11, 136	coe1f 22123	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ 𝐵 → (coe₁‘𝐺):ℕ₀⟶𝐾)
172	17, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (coe₁‘𝐺):ℕ₀⟶𝐾)
173	172	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝐺):ℕ₀⟶𝐾)
174	173, 106	ffvelcdmd 7089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐾)
175	136, 142	ringass 20186	. . . . . . . . . . . . . . . . . . . 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)))))
176	132, 174, 134, 141, 175	syl13anc 1370	. . . . . . . . . . . . . . . . . . 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)))))
177		ply1divex.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 = (1_r‘𝑅)
178	136, 142, 177	ringlidm 20198	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾) → ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))
179	132, 141, 178	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))
180	169, 176, 179	3eqtr3rd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) = (((coe₁‘𝐺)‘(𝐷‘𝐺)) · (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))))
181	162, 166, 180	3eqtr4rd 2778	. . . . . . . . . . . . . . . . 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)))
182	181	adantrr 716	. . . . . . . . . . . . . . . 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)))
183	10, 11, 13, 82, 102, 103, 104, 129, 153, 159, 135, 160, 182	deg1sublt 26039	. . . . . . . . . . . . . . 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))
184	183	adantlrr 720	. . . . . . . . . . . . . 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))
185		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (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)))))))
186	185	breq1d 5152	. . . . . . . . . . . . . . . 16 ⊢ (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)))
187		fvoveq1 7437	. . . . . . . . . . . . . . . . . 18 ⊢ (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))))
188	187	breq1d 5152	. . . . . . . . . . . . . . . . 17 ⊢ (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)))
189	188	rexbidv 3173	. . . . . . . . . . . . . . . 16 ⊢ (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)))
190	186, 189	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (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))))
191		simplrr 777	. . . . . . . . . . . . . . 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)))
192	81	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Grp)
193		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
194	13, 82	grpsubcl 18969	. . . . . . . . . . . . . . . . . 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)
195	192, 193, 152, 194	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝑔 − (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))) ∈ 𝐵)
196	195	adantrr 716	. . . . . . . . . . . . . . . 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)
197	196	adantlrr 720	. . . . . . . . . . . . . . 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)
198	190, 191, 197	rspcdva 3608	. . . . . . . . . . . . . 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)))
199	184, 198	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))
200	71	ad3antrrr 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Ring)
201		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵)
202	150	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ 𝐵)
203		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
204	13, 203	ringacl 20207	. . . . . . . . . . . . . . . . . 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)
205	200, 201, 202, 204	syl3anc 1369	. . . . . . . . . . . . . . . . 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)
206	81	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Grp)
207		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑔 ∈ 𝐵)
208	152	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ ((𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))( ·_𝑠 ‘𝑃)(𝑑(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ 𝐵)
209	17	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵)
210	13, 75	ringcl 20183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
211	200, 209, 201, 210	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
212	13, 203, 82	grpsubsub4 18982	. . . . . . . . . . . . . . . . . . . . . 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)))))))
213	206, 207, 208, 211, 212	syl13anc 1370	. . . . . . . . . . . . . . . . . . . . 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)))))))
214	13, 203, 75	ringdi 20193	. . . . . . . . . . . . . . . . . . . . . . 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))))))
215	200, 209, 201, 202, 214	syl13anc 1370	. . . . . . . . . . . . . . . . . . . . . 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))))))
216	215	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . 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)))))))
217	213, 216	eqtr4d 2770	. . . . . . . . . . . . . . . . . . . 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)))))))
218	217	fveq2d 6895	. . . . . . . . . . . . . . . . . . 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))))))))
219	218	breq1d 5152	. . . . . . . . . . . . . . . . . 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)))
220	219	biimpd 228	. . . . . . . . . . . . . . . . 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)))
221		oveq2 7422	. . . . . . . . . . . . . . . . . . . . 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))))))
222	221	oveq2d 7430	. . . . . . . . . . . . . . . . . . . 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)))))))
223	222	fveq2d 6895	. . . . . . . . . . . . . . . . . . 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))))))))
224	223	breq1d 5152	. . . . . . . . . . . . . . . . . 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)))
225	224	rspcev 3607	. . . . . . . . . . . . . . . . 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))
226	205, 220, 225	syl6an 683	. . . . . . . . . . . . . . . 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)))
227	226	rexlimdva 3150	. . . . . . . . . . . . . . 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)))
228	227	adantrr 716	. . . . . . . . . . . . . 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)))
229	228	adantlrr 720	. . . . . . . . . . . . 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)))
230	199, 229	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))
231	230	expr 456	. . . . . . . . . . 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)))
232	231	ralrimiva 3141	. . . . . . . . . 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)))
233		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝐷‘𝑔) = (𝐷‘𝑓))
234	233	breq1d 5152	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
235		fvoveq1 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))))
236	235	breq1d 5152	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
237	236	rexbidv 3173	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
238		oveq2 7422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑞 → (𝐺 ∙ 𝑟) = (𝐺 ∙ 𝑞))
239	238	oveq2d 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑞 → (𝑓 − (𝐺 ∙ 𝑟)) = (𝑓 − (𝐺 ∙ 𝑞)))
240	239	fveq2d 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑞 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))))
241	240	breq1d 5152	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
242	241	cbvrexvw 3230	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
243	237, 242	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
244	234, 243	imbi12d 344	. . . . . . . . . . 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	cbvralvw 3229	. . . . . . . . . 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	232, 245	sylib 217	. . . . . . . . 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 420	. . . . . . . 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	59, 64, 69, 64, 99, 249	nn0ind 12681	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
251	250	impcom 407	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
252	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝐹 ∈ 𝐵)
253	54, 251, 252	rspcdva 3608	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
254	253	rexlimdva 3150	. 2 ⊢ (𝜑 → (∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
255	48, 254	mpd 15	1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 ∪ cun 3942 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 -∞cmnf 11270 < clt 11272 ≤ cle 11273 − cmin 11468 ℕcn 12236 ℕ₀cn0 12496 ℤcz 12582 Basecbs 17173 +_gcplusg 17226 ._rcmulr 17227 ·_𝑠 cvsca 17230 0_gc0g 17414 Grpcgrp 18883 -_gcsg 18885 ._gcmg 19016 mulGrpcmgp 20067 1_rcur 20114 Ringcrg 20166 var₁cv1 22088 Poly₁cpl1 22089 coe₁cco1 22090 deg₁ cdg1 25980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-rlreg 21223 df-cnfld 21273 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mdeg 25981 df-deg1 25982

This theorem is referenced by: ply1divalg 26066

W3C validator