lmodprop2d - Intuitionistic Logic Explorer

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Theorem lmodprop2d 13438

Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 13439 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 13384	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐾 ∈ Grp)
2	1	a1i 9	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3		eqid 2177	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2177	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
5		eqid 2177	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
6		lmodprop2d.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝐾)
7		eqid 2177	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
8		eqid 2177	. . . . . 6 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
9		eqid 2177	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
10		eqid 2177	. . . . . 6 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
11	3, 4, 5, 6, 7, 8, 9, 10	islmod 13381	. . . . 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 1013	. . . 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 2256	. . . . . . 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 2256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
23	3, 6, 5, 7	lmodvscl 13395	. . . . . . 7 ⊢ ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
24	14, 18, 22, 23	syl3anc 1238	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
25	24, 21	eleqtrrd 2257	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
26	25	ralrimivva 2559	. . . 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 1178	. 2 ⊢ (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
29		lmodgrp 13384	. . . 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 12893	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
33	29, 32	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34		eqid 2177	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
35		eqid 2177	. . . . . 6 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
36		eqid 2177	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
37		lmodprop2d.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝐿)
38		eqid 2177	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
39		eqid 2177	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
40		eqid 2177	. . . . . 6 ⊢ (._r‘𝐺) = (._r‘𝐺)
41		eqid 2177	. . . . . 6 ⊢ (1_r‘𝐺) = (1_r‘𝐺)
42	34, 35, 36, 37, 38, 39, 40, 41	islmod 13381	. . . . 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 1013	. . . 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 13217	. . . 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 2256	. . . . . . 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 2256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
56	34, 37, 36, 38	lmodvscl 13395	. . . . . . 7 ⊢ ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·_𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
57	49, 52, 55, 56	syl3anc 1238	. . . . . 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 2261	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
61	60	ralrimivva 2559	. . . 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 1178	. 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 5902	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
70	66, 67, 68, 69	syl12anc 1236	. . . . . . . . . . . 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 2246	. . . . . . . . . . 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 1039	. . . . . . . . . . . . . . . 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 2256	. . . . . . . . . . . . . . . 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 2256	. . . . . . . . . . . . . . . 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 12890	. . . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑤(+_g‘𝐾)𝑧) = (𝑤(+_g‘𝐿)𝑧))
82	66, 68, 75, 81	syl12anc 1236	. . . . . . . . . . . . . 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 5891	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . 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 5901	. . . . . . . . . . . . . . 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 1237	. . . . . . . . . . . . . 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 5901	. . . . . . . . . . . . . . 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 1237	. . . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
93	66, 67, 75, 92	syl12anc 1236	. . . . . . . . . . . . . 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 5893	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 2192	. . . . . . . . . . 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 1040	. . . . . . . . . . . . . . . 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 2256	. . . . . . . . . . . . . . . 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 2256	. . . . . . . . . . . . . . . 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 13213	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+_g‘𝐹)𝑟) ∈ (Base‘𝐹))
103	97, 100, 101, 102	syl3anc 1238	. . . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . . 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 5890	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 5901	. . . . . . . . . . . . . . 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 1237	. . . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑞( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐿)𝑤))
116	66, 98, 68, 115	syl12anc 1236	. . . . . . . . . . . . . 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 5893	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 2192	. . . . . . . . . . 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 1312	. . . . . . . . . 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 13196	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(._r‘𝐹)𝑟) ∈ (Base‘𝐹))
122	97, 100, 101, 121	syl3anc 1238	. . . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5902	. . . . . . . . . . . . . . 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 5890	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 5902	. . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 5891	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 2192	. . . . . . . . . . 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 13203	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Ring → (1_r‘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))) → (1_r‘𝐹) ∈ (Base‘𝐹))
137	136, 99	eleqtrrd 2257	. . . . . . . . . . . . . 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)
138	58	oveqrspc2v 5902	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1_r‘𝐹) ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → ((1_r‘𝐹)( ·_𝑠 ‘𝐾)𝑤) = ((1_r‘𝐹)( ·_𝑠 ‘𝐿)𝑤))
139	66, 137, 68, 138	syl12anc 1236	. . . . . . . . . . . . 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))
140	16	3ad2ant1 1018	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝑃 = (Base‘𝐹))
141	44	3ad2ant1 1018	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝑃 = (Base‘𝐺))
142	46	3ad2antl1 1159	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(._r‘𝐹)𝑦) = (𝑥(._r‘𝐺)𝑦))
143		simp3 999	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝐹 ∈ Ring)
144	47	biimpa 296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐹 ∈ Ring) → 𝐺 ∈ Ring)
145	144	3adant2 1016	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → 𝐺 ∈ Ring)
146	140, 141, 142, 143, 145	rngidpropdg 13315	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ Grp ∧ 𝐹 ∈ Ring) → (1_r‘𝐹) = (1_r‘𝐺))
147	66, 72, 97, 146	syl3anc 1238	. . . . . . . . . . . . . 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))
148	147	oveq1d 5890	. . . . . . . . . . . . 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))
149	139, 148	eqtrd 2210	. . . . . . . . . . . 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))
150	149	eqeq1d 2186	. . . . . . . . . . 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))
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)) ∧ ((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)))
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)) ∧ ((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))))
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)) ∧ ((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))))
154	153	2ralbidva 2499	. . . . . . 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))))
155	154	2ralbidva 2499	. . . . . 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))))
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 2247	. . . . . . . . . . . 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 1316	. . . . . . . . . . 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)) ∧ ((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))))
161	157, 160	raleqbidv 2685	. . . . . . . . 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))))
162	157, 161	raleqbidv 2685	. . . . . . . 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))))
163	156, 162	raleqbidv 2685	. . . . . . 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))))
164	156, 163	raleqbidv 2685	. . . . . 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))))
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 2247	. . . . . . . . . . . 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 1316	. . . . . . . . . . 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)) ∧ ((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))))
170	166, 169	raleqbidv 2685	. . . . . . . . 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))))
171	166, 170	raleqbidv 2685	. . . . . . . 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))))
172	165, 171	raleqbidv 2685	. . . . . . 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))))
173	165, 172	raleqbidv 2685	. . . . . 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))))
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)) ∧ ((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))))
175	64, 65, 174	3anbi123d 1312	. . . 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)))))
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 705	1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5217 (class class class)co 5875 Basecbs 12462 +_gcplusg 12536 ._rcmulr 12537 Scalarcsca 12539 ·_𝑠 cvsca 12540 Grpcgrp 12877 1_rcur 13142 Ringcrg 13179 LModclmod 13377

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-pre-ltirr 7923 ax-pre-ltadd 7927

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-5 8981 df-6 8982 df-ndx 12465 df-slot 12466 df-base 12468 df-sets 12469 df-plusg 12549 df-mulr 12550 df-sca 12552 df-vsca 12553 df-0g 12707 df-mgm 12775 df-sgrp 12808 df-mnd 12818 df-grp 12880 df-mgp 13131 df-ur 13143 df-ring 13181 df-lmod 13379

This theorem is referenced by: lmodpropd 13439

W3C validator