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Theorem lmodprop2d 21161

Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 21162 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)

**Hypotheses**
Ref	Expression
lmodprop2d.b1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lmodprop2d.b2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lmodprop2d.f	⊢ 𝐹 = (Scalar‘𝐾)
lmodprop2d.g	⊢ 𝐺 = (Scalar‘𝐿)
lmodprop2d.p1	⊢ (𝜑 → 𝑃 = (Base‘𝐹))
lmodprop2d.p2	⊢ (𝜑 → 𝑃 = (Base‘𝐺))
lmodprop2d.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
lmodprop2d.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+_g‘𝐹)𝑦) = (𝑥(+_g‘𝐺)𝑦))
lmodprop2d.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(._r‘𝐹)𝑦) = (𝑥(._r‘𝐺)𝑦))
lmodprop2d.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))

**Assertion**
Ref	Expression
lmodprop2d	⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Distinct variable groups: 𝑥,𝑦,𝐵 𝑥,𝐹,𝑦 𝜑,𝑥,𝑦 𝑥,𝐺,𝑦 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝑥,𝑃,𝑦

Proof of Theorem lmodprop2d Dummy variables 𝑟 𝑞 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmodgrp 21104	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐾 ∈ Grp)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2760	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
5		eqid 2760	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
6		lmodprop2d.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝐾)
7		eqid 2760	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
8		eqid 2760	. . . . . 6 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
9		eqid 2760	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
10		eqid 2760	. . . . . 6 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
11	3, 4, 5, 6, 7, 8, 9, 10	islmod 21101	. . . . 5 ⊢ (𝐾 ∈ LMod ↔ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
12	11	simp2bi 1164	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐹 ∈ Ring)
13	12	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ LMod)
15		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
16		lmodprop2d.p1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
17	16	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
18	15, 17	eleqtrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐹))
19		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
20		lmodprop2d.b1	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
21	20	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
22	19, 21	eleqtrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
23	3, 6, 5, 7	lmodvscl 21115	. . . . . . 7 ⊢ ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
24	14, 18, 22, 23	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
25	24, 21	eleqtrrd 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
26	25	ralrimivva 3205	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
27	26	ex 418	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
28	2, 13, 27	3jcad 1147	. 2 ⊢ (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
29		lmodgrp 21104	. . . 4 ⊢ (𝐿 ∈ LMod → 𝐿 ∈ Grp)
30		lmodprop2d.b2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
31		lmodprop2d.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
32	20, 30, 31	grppropd 19124	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
33	29, 32	imbitrrid 249	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34		eqid 2760	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
35		eqid 2760	. . . . . 6 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
36		eqid 2760	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
37		lmodprop2d.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝐿)
38		eqid 2760	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
39		eqid 2760	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
40		eqid 2760	. . . . . 6 ⊢ (._r‘𝐺) = (._r‘𝐺)
41		eqid 2760	. . . . . 6 ⊢ (1_r‘𝐺) = (1_r‘𝐺)
42	34, 35, 36, 37, 38, 39, 40, 41	islmod 21101	. . . . 5 ⊢ (𝐿 ∈ LMod ↔ (𝐿 ∈ Grp ∧ 𝐺 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
43	42	simp2bi 1164	. . . 4 ⊢ (𝐿 ∈ LMod → 𝐺 ∈ Ring)
44		lmodprop2d.p2	. . . . 5 ⊢ (𝜑 → 𝑃 = (Base‘𝐺))
45		lmodprop2d.2	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+_g‘𝐹)𝑦) = (𝑥(+_g‘𝐺)𝑦))
46		lmodprop2d.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(._r‘𝐹)𝑦) = (𝑥(._r‘𝐺)𝑦))
47	16, 44, 45, 46	ringpropd 20481	. . . 4 ⊢ (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
48	43, 47	imbitrrid 249	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ LMod)
50		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
51	44	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
52	50, 51	eleqtrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐺))
53		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
54	30	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
55	53, 54	eleqtrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
56	34, 37, 36, 38	lmodvscl 21115	. . . . . . 7 ⊢ ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·_𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
57	49, 52, 55, 56	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
58		lmodprop2d.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
59	58	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
60	57, 59, 54	3eltr4d 2875	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
61	60	ralrimivva 3205	. . . 4 ⊢ ((𝜑 ∧ 𝐿 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
62	61	ex 418	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
63	33, 48, 62	3jcad 1147	. 2 ⊢ (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
64	32	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
65	47	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑)
67		simprlr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃)
68		simprrr 794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵)
69	58	oveqrspc2v 7435	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
70	66, 67, 68, 69	syl12anc 850	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
71	70	eleq1d 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵))
72		simplr1 1234	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Grp)
73	20	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐵 = (Base‘𝐾))
74	68, 73	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾))
75		simprrl 793	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ 𝐵)
76	75, 73	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾))
77	3, 4	grpcl 19114	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑤 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑤(+_g‘𝐾)𝑧) ∈ (Base‘𝐾))
78	72, 74, 76, 77	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑤(+_g‘𝐾)𝑧) ∈ (Base‘𝐾))
79	78, 73	eleqtrrd 2863	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑤(+_g‘𝐾)𝑧) ∈ 𝐵)
80	58	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ (𝑤(+_g‘𝐾)𝑧) ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐾)𝑧)))
81	66, 67, 79, 80	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐾)𝑧)))
82	31	oveqrspc2v 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑤(+_g‘𝐾)𝑧) = (𝑤(+_g‘𝐿)𝑧))
83	66, 68, 75, 82	syl12anc 850	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑤(+_g‘𝐾)𝑧) = (𝑤(+_g‘𝐿)𝑧))
84	83	oveq2d 7424	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐾)𝑧)) = (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)))
85	81, 84	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)))
86		simplr3 1236	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
87		ovrspc2v 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
88	67, 68, 86, 87	syl21anc 851	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
89		ovrspc2v 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵)
90	67, 75, 86, 89	syl21anc 851	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵)
91	31	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵)) → ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑧)))
92	66, 88, 90, 91	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑧)))
93	58	oveqrspc2v 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
94	66, 67, 75, 93	syl12anc 850	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
95	70, 94	oveq12d 7426	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)))
96	92, 95	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)))
97	85, 96	eqeq12d 2776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ↔ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧))))
98		simplr2 1235	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐹 ∈ Ring)
99		simprll 791	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑞 ∈ 𝑃)
100	16	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑃 = (Base‘𝐹))
101	99, 100	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑞 ∈ (Base‘𝐹))
102	67, 100	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘𝐹))
103	7, 8	ringacl 20469	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+_g‘𝐹)𝑟) ∈ (Base‘𝐹))
104	98, 101, 102, 103	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(+_g‘𝐹)𝑟) ∈ (Base‘𝐹))
105	104, 100	eleqtrrd 2863	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(+_g‘𝐹)𝑟) ∈ 𝑃)
106	58	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑞(+_g‘𝐹)𝑟) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤))
107	66, 105, 68, 106	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤))
108	45	oveqrspc2v 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(+_g‘𝐹)𝑟) = (𝑞(+_g‘𝐺)𝑟))
109	108	ad2ant2r 760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(+_g‘𝐹)𝑟) = (𝑞(+_g‘𝐺)𝑟))
110	109	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤))
111	107, 110	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤))
112		ovrspc2v 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑞( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
113	99, 68, 86, 112	syl21anc 851	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
114	31	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑞( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)) → ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
115	66, 113, 88, 114	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
116	58	oveqrspc2v 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑞( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐿)𝑤))
117	66, 99, 68, 116	syl12anc 850	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐿)𝑤))
118	117, 70	oveq12d 7426	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
119	115, 118	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
120	111, 119	eqeq12d 2776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ↔ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))))
121	71, 97, 120	3anbi123d 1464	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ↔ ((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))))
122	7, 9	ringcl 20439	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(._r‘𝐹)𝑟) ∈ (Base‘𝐹))
123	98, 101, 102, 122	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(._r‘𝐹)𝑟) ∈ (Base‘𝐹))
124	123, 100	eleqtrrd 2863	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(._r‘𝐹)𝑟) ∈ 𝑃)
125	58	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑞(._r‘𝐹)𝑟) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤))
126	66, 124, 68, 125	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤))
127	46	oveqrspc2v 7435	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(._r‘𝐹)𝑟) = (𝑞(._r‘𝐺)𝑟))
128	127	ad2ant2r 760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞(._r‘𝐹)𝑟) = (𝑞(._r‘𝐺)𝑟))
129	128	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤))
130	126, 129	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤))
131	58	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)) → (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
132	66, 99, 88, 131	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
133	70	oveq2d 7424	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
134	132, 133	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
135	130, 134	eqeq12d 2776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ↔ ((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))))
136	7, 10	ringidcl 20456	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Ring → (1_r‘𝐹) ∈ (Base‘𝐹))
137	98, 136	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1_r‘𝐹) ∈ (Base‘𝐹))
138	137, 100	eleqtrrd 2863	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1_r‘𝐹) ∈ 𝑃)
139	58	oveqrspc2v 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1_r‘𝐹) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = ((1_r‘𝐹)( ·_𝑠 ‘𝐿)𝑤))
140	66, 138, 68, 139	syl12anc 850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = ((1_r‘𝐹)( ·_𝑠 ‘𝐿)𝑤))
141	16, 44, 46	rngidpropd 20607	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝐹) = (1_r‘𝐺))
142	141	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1_r‘𝐹) = (1_r‘𝐺))
143	142	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1_r‘𝐹)( ·_𝑠 ‘𝐿)𝑤) = ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤))
144	140, 143	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤))
145	144	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤 ↔ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))
146	135, 145	anbi12d 644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤) ↔ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)))
147	121, 146	anbi12d 644	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
148	147	anassrs 473	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
149	148	2ralbidva 3224	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
150	149	2ralbidva 3224	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑞 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
151	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
152	20	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
153	152	eleq2d 2846	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ↔ (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾)))
154	153	3anbi1d 1468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ↔ ((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)))))
155	154	anbi1d 643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → ((((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
156	152, 155	raleqbidv 3334	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
157	152, 156	raleqbidv 3334	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
158	151, 157	raleqbidv 3334	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
159	151, 158	raleqbidv 3334	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑞 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))))
160	44	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
161	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
162	161	eleq2d 2846	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → ((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ↔ (𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿)))
163	162	3anbi1d 1468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ↔ ((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))))
164	163	anbi1d 643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → ((((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)) ↔ (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
165	161, 164	raleqbidv 3334	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
166	161, 165	raleqbidv 3334	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)) ↔ ∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
167	160, 166	raleqbidv 3334	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)) ↔ ∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
168	160, 167	raleqbidv 3334	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑞 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
169	150, 159, 168	3bitr3d 312	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤))))
170	64, 65, 169	3anbi123d 1464	. . . 4 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾) ∧ (𝑟( ·_𝑠 ‘𝐾)(𝑤(+_g‘𝐾)𝑧)) = ((𝑟( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑧)) ∧ ((𝑞(+_g‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤))) ∧ (((𝑞(._r‘𝐹)𝑟)( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) ∧ ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = 𝑤))) ↔ (𝐿 ∈ Grp ∧ 𝐺 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐺)∀𝑟 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑤) ∈ (Base‘𝐿) ∧ (𝑟( ·_𝑠 ‘𝐿)(𝑤(+_g‘𝐿)𝑧)) = ((𝑟( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑧)) ∧ ((𝑞(+_g‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤))) ∧ (((𝑞(._r‘𝐺)𝑟)( ·_𝑠 ‘𝐿)𝑤) = (𝑞( ·_𝑠 ‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) ∧ ((1_r‘𝐺)( ·_𝑠 ‘𝐿)𝑤) = 𝑤)))))
171	170, 11, 42	3bitr4g 317	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
172	171	ex 418	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)))
173	28, 63, 172	pm5.21ndd 382	1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ‘cfv 6527 (class class class)co 7408 Basecbs 17349 +_gcplusg 17390 ._rcmulr 17391 Scalarcsca 17393 ·_𝑠 cvsca 17394 Grpcgrp 19106 1_rcur 20369 Ringcrg 20421 LModclmod 21097

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-sets 17304 df-slot 17322 df-ndx 17334 df-base 17350 df-plusg 17403 df-0g 17574 df-mgm 18778 df-sgrp 18870 df-mnd 18886 df-grp 19109 df-mgp 20323 df-ur 20370 df-ring 20423 df-lmod 21099

This theorem is used by: lmodpropd 21162 lvecprop2d 21406

W3C validator