Theorem lmodprop2d 14320

Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 14321 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 14266	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐾 ∈ Grp)
2	1	a1i 9	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3		eqid 2229	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2229	. . . . . 6 ⊢ (+g‘𝐾) = (+g‘𝐾)
5		eqid 2229	. . . . . 6 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
6		lmodprop2d.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝐾)
7		eqid 2229	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
8		eqid 2229	. . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹)
9		eqid 2229	. . . . . 6 ⊢ (.r‘𝐹) = (.r‘𝐹)
10		eqid 2229	. . . . . 6 ⊢ (1r‘𝐹) = (1r‘𝐹)
11	3, 4, 5, 6, 7, 8, 9, 10	islmod 14263	. . . . 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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
12	11	simp2bi 1037	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐹 ∈ Ring)
13	12	a1i 9	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ LMod)
15		simprl 529	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
16		lmodprop2d.p1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
17	16	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
18	15, 17	eleqtrd 2308	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐹))
19		simprr 531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
20		lmodprop2d.b1	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
21	20	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
22	19, 21	eleqtrd 2308	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
23	3, 6, 5, 7	lmodvscl 14277	. . . . . . 7 ⊢ ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
24	14, 18, 22, 23	syl3anc 1271	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
25	24, 21	eleqtrrd 2309	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)
26	25	ralrimivva 2612	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)
27	26	ex 115	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵))
28	2, 13, 27	3jcad 1202	. 2 ⊢ (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
29		lmodgrp 14266	. . . 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 13558	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
33	29, 32	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34		eqid 2229	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
35		eqid 2229	. . . . . 6 ⊢ (+g‘𝐿) = (+g‘𝐿)
36		eqid 2229	. . . . . 6 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
37		lmodprop2d.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝐿)
38		eqid 2229	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
39		eqid 2229	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
40		eqid 2229	. . . . . 6 ⊢ (.r‘𝐺) = (.r‘𝐺)
41		eqid 2229	. . . . . 6 ⊢ (1r‘𝐺) = (1r‘𝐺)
42	34, 35, 36, 37, 38, 39, 40, 41	islmod 14263	. . . . 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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
43	42	simp2bi 1037	. . . 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 14009	. . . 4 ⊢ (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
48	43, 47	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ LMod)
50		simprl 529	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
51	44	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
52	50, 51	eleqtrd 2308	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐺))
53		simprr 531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
54	30	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
55	53, 54	eleqtrd 2308	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
56	34, 37, 36, 38	lmodvscl 14277	. . . . . . 7 ⊢ ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
57	49, 52, 55, 56	syl3anc 1271	. . . . . 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 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
60	57, 59, 54	3eltr4d 2313	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)
61	60	ralrimivva 2612	. . . 4 ⊢ ((𝜑 ∧ 𝐿 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)
62	61	ex 115	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵))
63	33, 48, 62	3jcad 1202	. 2 ⊢ (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
64	32	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
65	47	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66		simpll 527	. . . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 540	. . . . . . . . . . . . 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 6034	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑤) = (𝑟( ·𝑠 ‘𝐿)𝑤))
70	66, 67, 68, 69	syl12anc 1269	. . . . . . . . . . . 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 2298	. . . . . . . . . . 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 1063	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . . 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 539	. . . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . . 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, 72, 74, 76	grpcld 13555	. . . . . . . . . . . . . . 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))
78	77, 73	eleqtrrd 2309	. . . . . . . . . . . . . 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)
79	58	oveqrspc2v 6034	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ (𝑤(+g‘𝐾)𝑧) ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)(𝑤(+g‘𝐾)𝑧)) = (𝑟( ·𝑠 ‘𝐿)(𝑤(+g‘𝐾)𝑧)))
80	66, 67, 78, 79	syl12anc 1269	. . . . . . . . . . . . 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)))
81	31	oveqrspc2v 6034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝐿)𝑧))
82	66, 68, 75, 81	syl12anc 1269	. . . . . . . . . . . . . 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))
83	82	oveq2d 6023	. . . . . . . . . . . . 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)))
84	80, 83	eqtrd 2262	. . . . . . . . . . . 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)))
85		simplr3 1065	. . . . . . . . . . . . . . 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)
86		ovrspc2v 6033	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ 𝐵)
87	67, 68, 85, 86	syl21anc 1270	. . . . . . . . . . . . . 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)
88		ovrspc2v 6033	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ 𝐵)
89	67, 75, 85, 88	syl21anc 1270	. . . . . . . . . . . . . 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)
90	31	oveqrspc2v 6034	. . . . . . . . . . . . . 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)))
91	66, 87, 89, 90	syl12anc 1269	. . . . . . . . . . . . 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)))
92	58	oveqrspc2v 6034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑧) = (𝑟( ·𝑠 ‘𝐿)𝑧))
93	66, 67, 75, 92	syl12anc 1269	. . . . . . . . . . . . . 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))
94	70, 93	oveq12d 6025	. . . . . . . . . . . . 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)))
95	91, 94	eqtrd 2262	. . . . . . . . . . . 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)))
96	84, 95	eqeq12d 2244	. . . . . . . . . . 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))))
97		simplr2 1064	. . . . . . . . . . . . . . . 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)
98		simprll 537	. . . . . . . . . . . . . . . . 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)
99	16	ad2antrr 488	. . . . . . . . . . . . . . . . 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))
100	98, 99	eleqtrd 2308	. . . . . . . . . . . . . . . 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))
101	67, 99	eleqtrd 2308	. . . . . . . . . . . . . . . 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))
102	7, 8	ringacl 14001	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+g‘𝐹)𝑟) ∈ (Base‘𝐹))
103	97, 100, 101, 102	syl3anc 1271	. . . . . . . . . . . . . . 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))
104	103, 99	eleqtrrd 2309	. . . . . . . . . . . . . 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)
105	58	oveqrspc2v 6034	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑞(+g‘𝐹)𝑟) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((𝑞(+g‘𝐹)𝑟)( ·𝑠 ‘𝐾)𝑤) = ((𝑞(+g‘𝐹)𝑟)( ·𝑠 ‘𝐿)𝑤))
106	66, 104, 68, 105	syl12anc 1269	. . . . . . . . . . . . 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))
107	45	oveqrspc2v 6034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(+g‘𝐹)𝑟) = (𝑞(+g‘𝐺)𝑟))
108	107	ad2ant2r 509	. . . . . . . . . . . . . 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))
109	108	oveq1d 6022	. . . . . . . . . . . . 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))
110	106, 109	eqtrd 2262	. . . . . . . . . . . 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))
111		ovrspc2v 6033	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝑞( ·𝑠 ‘𝐾)𝑤) ∈ 𝐵)
112	98, 68, 85, 111	syl21anc 1270	. . . . . . . . . . . . . 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)
113	31	oveqrspc2v 6034	. . . . . . . . . . . . . 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)))
114	66, 112, 87, 113	syl12anc 1269	. . . . . . . . . . . . 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)))
115	58	oveqrspc2v 6034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑞( ·𝑠 ‘𝐾)𝑤) = (𝑞( ·𝑠 ‘𝐿)𝑤))
116	66, 98, 68, 115	syl12anc 1269	. . . . . . . . . . . . . 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))
117	116, 70	oveq12d 6025	. . . . . . . . . . . . 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)))
118	114, 117	eqtrd 2262	. . . . . . . . . . . 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)))
119	110, 118	eqeq12d 2244	. . . . . . . . . . 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))))
120	71, 96, 119	3anbi123d 1346	. . . . . . . . . 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)))))
121	7, 9	ringcl 13984	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(.r‘𝐹)𝑟) ∈ (Base‘𝐹))
122	97, 100, 101, 121	syl3anc 1271	. . . . . . . . . . . . . . 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))
123	122, 99	eleqtrrd 2309	. . . . . . . . . . . . . 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)
124	58	oveqrspc2v 6034	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑞(.r‘𝐹)𝑟) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((𝑞(.r‘𝐹)𝑟)( ·𝑠 ‘𝐾)𝑤) = ((𝑞(.r‘𝐹)𝑟)( ·𝑠 ‘𝐿)𝑤))
125	66, 123, 68, 124	syl12anc 1269	. . . . . . . . . . . . 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))
126	46	oveqrspc2v 6034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(.r‘𝐹)𝑟) = (𝑞(.r‘𝐺)𝑟))
127	126	ad2ant2r 509	. . . . . . . . . . . . . 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))
128	127	oveq1d 6022	. . . . . . . . . . . . 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))
129	125, 128	eqtrd 2262	. . . . . . . . . . . 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))
130	58	oveqrspc2v 6034	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ 𝐵)) → (𝑞( ·𝑠 ‘𝐾)(𝑟( ·𝑠 ‘𝐾)𝑤)) = (𝑞( ·𝑠 ‘𝐿)(𝑟( ·𝑠 ‘𝐾)𝑤)))
131	66, 98, 87, 130	syl12anc 1269	. . . . . . . . . . . . 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)))
132	70	oveq2d 6023	. . . . . . . . . . . . 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)))
133	131, 132	eqtrd 2262	. . . . . . . . . . . 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)))
134	129, 133	eqeq12d 2244	. . . . . . . . . . 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))))
135	7, 10	ringidcl 13991	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹))
136	97, 135	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1r‘𝐹) ∈ (Base‘𝐹))
137	136, 99	eleqtrrd 2309	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1r‘𝐹) ∈ 𝑃)
138	58	oveqrspc2v 6034	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1r‘𝐹) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = ((1r‘𝐹)( ·𝑠 ‘𝐿)𝑤))
139	66, 137, 68, 138	syl12anc 1269	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = ((1r‘𝐹)( ·𝑠 ‘𝐿)𝑤))
140	16	3ad2ant1 1042	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝑃 = (Base‘𝐹))
141	44	3ad2ant1 1042	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝑃 = (Base‘𝐺))
142	46	3ad2antl1 1183	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦))
143		simp3 1023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝐹 ∈ Ring)
144	47	biimpa 296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐹 ∈ Ring) → 𝐺 ∈ Ring)
145	144	3adant2 1040	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝐺 ∈ Ring)
146	140, 141, 142, 143, 145	rngidpropdg 14118	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → (1r‘𝐹) = (1r‘𝐺))
147	66, 72, 97, 146	syl3anc 1271	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (1r‘𝐹) = (1r‘𝐺))
148	147	oveq1d 6022	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1r‘𝐹)( ·𝑠 ‘𝐿)𝑤) = ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤))
149	139, 148	eqtrd 2262	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤))
150	149	eqeq1d 2238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤 ↔ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))
151	134, 150	anbi12d 473	. . . . . . . . . 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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤) ↔ (((𝑞(.r‘𝐺)𝑟)( ·𝑠 ‘𝐿)𝑤) = (𝑞( ·𝑠 ‘𝐿)(𝑟( ·𝑠 ‘𝐿)𝑤)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤)))
152	120, 151	anbi12d 473	. . . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
153	152	anassrs 400	. . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
154	153	2ralbidva 2552	. . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
155	154	2ralbidva 2552	. . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
156	16	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
157	20	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
158	157	eleq2d 2299	. . . . . . . . . . . 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)))
159	158	3anbi1d 1350	. . . . . . . . . . 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)))))
160	159	anbi1d 465	. . . . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
161	157, 160	raleqbidv 2744	. . . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
162	157, 161	raleqbidv 2744	. . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
163	156, 162	raleqbidv 2744	. . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
164	156, 163	raleqbidv 2744	. . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝐾)𝑤) = 𝑤))))
165	44	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
166	30	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
167	166	eleq2d 2299	. . . . . . . . . . . 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)))
168	167	3anbi1d 1350	. . . . . . . . . . 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)))))
169	168	anbi1d 465	. . . . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
170	166, 169	raleqbidv 2744	. . . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
171	166, 170	raleqbidv 2744	. . . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
172	165, 171	raleqbidv 2744	. . . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
173	165, 172	raleqbidv 2744	. . . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
174	155, 164, 173	3bitr3d 218	. . . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤))))
175	64, 65, 174	3anbi123d 1346	. . . 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)) ∧ ((1r‘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)) ∧ ((1r‘𝐺)( ·𝑠 ‘𝐿)𝑤) = 𝑤)))))
176	175, 11, 42	3bitr4g 223	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
177	176	ex 115	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)))
178	28, 63, 177	pm5.21ndd 710	1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  +_gcplusg 13118  ._rcmulr 13119  Scalarcsca 13121   ·_𝑠 cvsca 13122  Grpcgrp 13541  1_rcur 13930  Ringcrg 13967  LModclmod 14259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-plusg 13131  df-mulr 13132  df-sca 13134  df-vsca 13135  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-mgp 13892  df-ur 13931  df-ring 13969  df-lmod 14261

This theorem is referenced by:  lmodpropd  14321