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Theorem ply1divex 26100

Description: Lemma for ply1divalg 26101: 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 6902	. . . . 5 ⊢ (𝐹 = 0 → (𝐷‘𝐹) = (𝐷‘ 0 ))
2	1	breq1d 5162	. . . 4 ⊢ (𝐹 = 0 → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
3	2	rexbidv 3176	. . 3 ⊢ (𝐹 = 0 → (∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) ↔ ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑)))
4		nnssnn0 12515	. . . . 5 ⊢ ℕ ⊆ ℕ₀
5		ply1divalg.r1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	5	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Ring)
7		ply1divalg.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
8	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐵)
9		simpr 483	. . . . . . . . 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 26052	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
15	6, 8, 9, 14	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
16	15	nn0red 12573	. . . . . . 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 26052	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
20	5, 17, 18, 19	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
21	20	nn0red 12573	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ)
22	21	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐺) ∈ ℝ)
23	16, 22	resubcld 11682	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ≠ 0 ) → ((𝐷‘𝐹) − (𝐷‘𝐺)) ∈ ℝ)
24		arch 12509	. . . . . 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 4051	. . . . 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 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐹) ∈ ℝ)
29	21	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℝ)
30		nn0re 12521	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℝ)
31	30	adantl 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ 0 ) ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℝ)
32	28, 29, 31	ltsubadd2d 11852	. . . . . 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 3165	. . . 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 12527	. . . 4 ⊢ 0 ∈ ℕ₀
37	10, 11, 12	deg1z 26051	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞)
38	5, 37	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷‘ 0 ) = -∞)
39		0re 11256	. . . . . . 7 ⊢ 0 ∈ ℝ
40		readdcl 11231	. . . . . . 7 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐷‘𝐺) + 0) ∈ ℝ)
41	21, 39, 40	sylancl 584	. . . . . 6 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) ∈ ℝ)
42	41	mnfltd 13146	. . . . 5 ⊢ (𝜑 → -∞ < ((𝐷‘𝐺) + 0))
43	38, 42	eqbrtrd 5174	. . . 4 ⊢ (𝜑 → (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0))
44		oveq2 7434	. . . . . 6 ⊢ (𝑑 = 0 → ((𝐷‘𝐺) + 𝑑) = ((𝐷‘𝐺) + 0))
45	44	breq2d 5164	. . . . 5 ⊢ (𝑑 = 0 → ((𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)))
46	45	rspcev 3611	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 0)) → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
47	36, 43, 46	sylancr 585	. . 3 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘ 0 ) < ((𝐷‘𝐺) + 𝑑))
48	3, 35, 47	pm2.61ne 3024	. 2 ⊢ (𝜑 → ∃𝑑 ∈ ℕ₀ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑))
49		fveq2 6902	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
50	49	breq1d 5162	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) ↔ (𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑)))
51		fvoveq1 7449	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))))
52	51	breq1d 5162	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
53	52	rexbidv 3176	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
54	50, 53	imbi12d 343	. . . 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 7434	. . . . . . . . . 10 ⊢ (𝑎 = 0 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 0))
56	55	breq2d 5164	. . . . . . . . 9 ⊢ (𝑎 = 0 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)))
57	56	imbi1d 340	. . . . . . . 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 3175	. . . . . . 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 339	. . . . . 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 7434	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + 𝑑))
61	60	breq2d 5164	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑)))
62	61	imbi1d 340	. . . . . . . 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 3175	. . . . . . 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 339	. . . . . 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 7434	. . . . . . . . . 10 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝐺) + 𝑎) = ((𝐷‘𝐺) + (𝑑 + 1)))
66	65	breq2d 5164	. . . . . . . . 9 ⊢ (𝑎 = (𝑑 + 1) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑎) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
67	66	imbi1d 340	. . . . . . . 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 3175	. . . . . . 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 339	. . . . . 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 22185	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
71	5, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ Ring)
72	13, 12	ring0cl 20217	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 0 ∈ 𝐵)
73	71, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
74	73	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → 0 ∈ 𝐵)
75		ply1divalg.t	. . . . . . . . . . . . . . . . 17 ⊢ ∙ = (._r‘𝑃)
76	13, 75, 12	ringrz 20244	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∙ 0 ) = 0 )
77	71, 17, 76	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ∙ 0 ) = 0 )
78	77	oveq2d 7442	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
79	78	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − (𝐺 ∙ 0 )) = (𝑓 − 0 ))
80		ringgrp 20192	. . . . . . . . . . . . . . 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 18995	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ Grp ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
84	81, 83	sylan 578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 − 0 ) = 𝑓)
85	79, 84	eqtr2d 2769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 = (𝑓 − (𝐺 ∙ 0 )))
86	85	fveq2d 6906	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐷‘𝑓) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
87	20	nn0cnd 12574	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℂ)
88	87	addridd 11454	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
89	88	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝐺) + 0) = (𝐷‘𝐺))
90	86, 89	breq12d 5165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
91	90	biimpa 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺))
92		oveq2 7434	. . . . . . . . . . . . 13 ⊢ (𝑞 = 0 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 0 ))
93	92	oveq2d 7442	. . . . . . . . . . . 12 ⊢ (𝑞 = 0 → (𝑓 − (𝐺 ∙ 𝑞)) = (𝑓 − (𝐺 ∙ 0 )))
94	93	fveq2d 6906	. . . . . . . . . . 11 ⊢ (𝑞 = 0 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝑓 − (𝐺 ∙ 0 ))))
95	94	breq1d 5162	. . . . . . . . . 10 ⊢ (𝑞 = 0 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)))
96	95	rspcev 3611	. . . . . . . . 9 ⊢ (( 0 ∈ 𝐵 ∧ (𝐷‘(𝑓 − (𝐺 ∙ 0 ))) < (𝐷‘𝐺)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
97	74, 91, 96	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ (𝐷‘𝑓) < ((𝐷‘𝐺) + 0)) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
98	97	ex 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
99	98	ralrimiva 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 0) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
100		nn0addcl 12547	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷‘𝐺) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
101	20, 100	sylan 578	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
102	101	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
103	5	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑅 ∈ Ring)
104		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → 𝑔 ∈ 𝐵)
105	10, 11, 13	deg1cl 26047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 ∈ 𝐵 → (𝐷‘𝑔) ∈ (ℕ₀ ∪ {-∞}))
106	20	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℕ₀)
107		peano2nn0 12552	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
108	107	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝑑 + 1) ∈ ℕ₀)
109	106, 108	nn0addcld 12576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℕ₀)
110	109	nn0zd 12624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + (𝑑 + 1)) ∈ ℤ)
111		degltlem1 26036	. . . . . . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ (((𝐷‘𝐺) + (𝑑 + 1)) − 1))
115	20	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℕ₀)
116	115	nn0cnd 12574	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝐷‘𝐺) ∈ ℂ)
117		nn0cn 12522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℂ)
118	117	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝑑 ∈ ℂ)
119		peano2cn 11426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ℂ → (𝑑 + 1) ∈ ℂ)
120	118, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑑 + 1) ∈ ℂ)
121		1cnd 11249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 1 ∈ ℂ)
122	116, 120, 121	addsubassd 11631	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + ((𝑑 + 1) − 1)))
123		ax-1cn 11206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
124		pncan 11506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑑 + 1) − 1) = 𝑑)
125	118, 123, 124	sylancl 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑑 + 1) − 1) = 𝑑)
126	125	oveq2d 7442	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐺) + ((𝑑 + 1) − 1)) = ((𝐷‘𝐺) + 𝑑))
127	122, 126	eqtrd 2768	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
128	127	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (((𝐷‘𝐺) + (𝑑 + 1)) − 1) = ((𝐷‘𝐺) + 𝑑))
129	114, 128	breqtrd 5178	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑔 ∈ 𝐵 ∧ (𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)))) → (𝐷‘𝑔) ≤ ((𝐷‘𝐺) + 𝑑))
130	71	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Ring)
131	17	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐺 ∈ 𝐵)
132	5	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑅 ∈ Ring)
133		ply1divex.i	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ 𝐾)
134	133	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ 𝐾)
135		eqid 2728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑔) = (coe₁‘𝑔)
136		ply1divex.k	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐾 = (Base‘𝑅)
137	135, 13, 11, 136	coe1f 22149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 ∈ 𝐵 → (coe₁‘𝑔):ℕ₀⟶𝐾)
138	137	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝑔):ℕ₀⟶𝐾)
139		simplr 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℕ₀)
140	106, 139	nn0addcld 12576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) ∈ ℕ₀)
141	138, 140	ffvelcdmd 7100	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾)
142		ply1divex.u	. . . . . . . . . . . . . . . . . . . . 21 ⊢ · = (._r‘𝑅)
143	136, 142	ringcl 20204	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐾 ∧ ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)) ∈ 𝐾) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
144	132, 134, 141, 143	syl3anc 1368	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐼 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) ∈ 𝐾)
145		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
146		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
147		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
148		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
149	136, 11, 145, 146, 147, 148, 13	ply1tmcl 22210	. . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . 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 20204	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 12573	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℝ)
155	154	leidd 11820	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ≤ (𝐷‘𝐺))
156	10, 136, 11, 145, 146, 147, 148	deg1tmle 26081	. . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . 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 26073	. . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
162	161, 136, 11, 145, 146, 147, 148, 13, 75, 142, 131, 132, 144, 139, 106	coe1tmmul2fv 22216	. . . . . . . . . . . . . . . . . 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 12574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (𝐷‘𝐺) ∈ ℂ)
164	117	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑑 ∈ ℂ)
165	163, 164	addcomd 11456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((𝐷‘𝐺) + 𝑑) = (𝑑 + (𝐷‘𝐺)))
166	165	fveq2d 6906	. . . . . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
169	168	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((((coe₁‘𝐺)‘(𝐷‘𝐺)) · 𝐼) · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))))
170		eqid 2728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
171	170, 13, 11, 136	coe1f 22149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ 𝐵 → (coe₁‘𝐺):ℕ₀⟶𝐾)
172	17, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (coe₁‘𝐺):ℕ₀⟶𝐾)
173	172	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → (coe₁‘𝐺):ℕ₀⟶𝐾)
174	173, 106	ffvelcdmd 7100	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐾)
175	136, 142	ringass 20207	. . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . 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 20219	. . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → ( 1 · ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑))) = ((coe₁‘𝑔)‘((𝐷‘𝐺) + 𝑑)))
180	169, 176, 179	3eqtr3rd 2777	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 26074	. . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑃 ∈ Grp)
193		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
194	13, 82	grpsubcl 18990	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . 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 3612	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Ring)
201		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵)
202	150	adantr 479	. . . . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . . 19 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
204	13, 203	ringacl 20228	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Grp)
207		simplr 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝑔 ∈ 𝐵)
208	152	adantr 479	. . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵)
210	13, 75	ringcl 20204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
211	200, 209, 201, 210	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑔 ∈ 𝐵) ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
212	13, 203, 82	grpsubsub4 19003	. . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . 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 20214	. . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . . . . 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 3611	. . . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . 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 719	. . . . . . . . . . . . 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 455	. . . . . . . . . . 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 3143	. . . . . . . . . 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 6902	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝐷‘𝑔) = (𝐷‘𝑓))
234	233	breq1d 5162	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝐷‘𝑔) < ((𝐷‘𝐺) + (𝑑 + 1)) ↔ (𝐷‘𝑓) < ((𝐷‘𝐺) + (𝑑 + 1))))
235		fvoveq1 7449	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))))
236	235	breq1d 5162	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
237	236	rexbidv 3176	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
238		oveq2 7434	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑞 → (𝐺 ∙ 𝑟) = (𝐺 ∙ 𝑞))
239	238	oveq2d 7442	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑞 → (𝑓 − (𝐺 ∙ 𝑟)) = (𝑓 − (𝐺 ∙ 𝑞)))
240	239	fveq2d 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑞 → (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) = (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))))
241	240	breq1d 5162	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 → ((𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
242	241	cbvrexvw 3233	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
243	237, 242	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∃𝑟 ∈ 𝐵 (𝐷‘(𝑔 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺) ↔ ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
244	234, 243	imbi12d 343	. . . . . . . . . . 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 3232	. . . . . . . . . 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 419	. . . . . . . 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 12697	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))))
251	250	impcom 406	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
252	7	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → 𝐹 ∈ 𝐵)
253	54, 251, 252	rspcdva 3612	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝐷‘𝐹) < ((𝐷‘𝐺) + 𝑑) → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)))
254	253	rexlimdva 3152	. 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∪ cun 3947 ⊆ wss 3949 {csn 4632 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℂcc 11146 ℝcr 11147 0cc0 11148 1c1 11149 + caddc 11151 -∞cmnf 11286 < clt 11288 ≤ cle 11289 − cmin 11484 ℕcn 12252 ℕ₀cn0 12512 ℤcz 12598 Basecbs 17189 +_gcplusg 17242 ._rcmulr 17243 ·_𝑠 cvsca 17246 0_gc0g 17430 Grpcgrp 18904 -_gcsg 18906 ._gcmg 19037 mulGrpcmgp 20088 1_rcur 20135 Ringcrg 20187 var₁cv1 22113 Poly₁cpl1 22114 coe₁cco1 22115 deg₁ cdg1 26015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-rlreg 21244 df-cnfld 21294 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-psr1 22117 df-vr1 22118 df-ply1 22119 df-coe1 22120 df-mdeg 26016 df-deg1 26017

This theorem is referenced by: ply1divalg 26101

W3C validator