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Theorem lmodprop2d 18925

Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 18926 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 18870	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐾 ∈ Grp)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3		eqid 2622	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2622	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
5		eqid 2622	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
6		lmodprop2d.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝐾)
7		eqid 2622	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
8		eqid 2622	. . . . . 6 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
9		eqid 2622	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
10		eqid 2622	. . . . . 6 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
11	3, 4, 5, 6, 7, 8, 9, 10	islmod 18867	. . . . 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 1077	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐹 ∈ Ring)
13	12	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ LMod)
15		simprl 794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
16		lmodprop2d.p1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
17	16	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
18	15, 17	eleqtrd 2703	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐹))
19		simprr 796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
20		lmodprop2d.b1	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
21	20	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
22	19, 21	eleqtrd 2703	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
23	3, 6, 5, 7	lmodvscl 18880	. . . . . . 7 ⊢ ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
24	14, 18, 22, 23	syl3anc 1326	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
25	24, 21	eleqtrrd 2704	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
26	25	ralrimivva 2971	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
27	26	ex 450	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
28	2, 13, 27	3jcad 1243	. 2 ⊢ (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
29		lmodgrp 18870	. . . 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 17437	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
33	29, 32	syl5ibr 236	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34		eqid 2622	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
35		eqid 2622	. . . . . 6 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
36		eqid 2622	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
37		lmodprop2d.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝐿)
38		eqid 2622	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
39		eqid 2622	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
40		eqid 2622	. . . . . 6 ⊢ (._r‘𝐺) = (._r‘𝐺)
41		eqid 2622	. . . . . 6 ⊢ (1_r‘𝐺) = (1_r‘𝐺)
42	34, 35, 36, 37, 38, 39, 40, 41	islmod 18867	. . . . 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 1077	. . . 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 18582	. . . 4 ⊢ (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
48	43, 47	syl5ibr 236	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ LMod)
50		simprl 794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
51	44	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
52	50, 51	eleqtrd 2703	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐺))
53		simprr 796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
54	30	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
55	53, 54	eleqtrd 2703	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
56	34, 37, 36, 38	lmodvscl 18880	. . . . . . 7 ⊢ ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·_𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
57	49, 52, 55, 56	syl3anc 1326	. . . . . 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 751	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
60	57, 59, 54	3eltr4d 2716	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
61	60	ralrimivva 2971	. . . 4 ⊢ ((𝜑 ∧ 𝐿 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
62	61	ex 450	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
63	33, 48, 62	3jcad 1243	. 2 ⊢ (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
64	32	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
65	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66		simpll 790	. . . . . . . . . . . . 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 803	. . . . . . . . . . . . 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 805	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
70	66, 67, 68, 69	syl12anc 1324	. . . . . . . . . . . 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 2686	. . . . . . . . . . 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 1103	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . 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 2703	. . . . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . . . . 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 2703	. . . . . . . . . . . . . . . 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 17430	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑤 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑤(+_g‘𝐾)𝑧) ∈ (Base‘𝐾))
78	72, 74, 76, 77	syl3anc 1326	. . . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑤(+_g‘𝐾)𝑧) = (𝑤(+_g‘𝐿)𝑧))
83	66, 68, 75, 82	syl12anc 1324	. . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 1105	. . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . 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 1325	. . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . 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 1325	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
94	66, 67, 75, 93	syl12anc 1324	. . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 2637	. . . . . . . . . . 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 1104	. . . . . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . 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 2703	. . . . . . . . . . . . . . . 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 2703	. . . . . . . . . . . . . . . 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 18578	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+_g‘𝐹)𝑟) ∈ (Base‘𝐹))
104	98, 101, 102, 103	syl3anc 1326	. . . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(+_g‘𝐹)𝑟) = (𝑞(+_g‘𝐺)𝑟))
109	108	ad2ant2r 783	. . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . 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 1325	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑞( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐿)𝑤))
117	66, 99, 68, 116	syl12anc 1324	. . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 2637	. . . . . . . . . . 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 1399	. . . . . . . . . 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 18561	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(._r‘𝐹)𝑟) ∈ (Base‘𝐹))
123	98, 101, 102, 122	syl3anc 1326	. . . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(._r‘𝐹)𝑟) = (𝑞(._r‘𝐺)𝑟))
128	127	ad2ant2r 783	. . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 6666	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 2637	. . . . . . . . . . 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 18568	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Ring → (1_r‘𝐹) ∈ (Base‘𝐹))
137	98, 136	syl 17	. . . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 18695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝐹) = (1_r‘𝐺))
142	141	ad2antrr 762	. . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 2624	. . . . . . . . . . 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 747	. . . . . . . . . 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 747	. . . . . . . . 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 680	. . . . . . . 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 2988	. . . . . . 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 2988	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
152	20	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
153	152	eleq2d 2687	. . . . . . . . . . . 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 1403	. . . . . . . . . . 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 741	. . . . . . . . . 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 3152	. . . . . . . . 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 3152	. . . . . . . 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 3152	. . . . . . 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 3152	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
161	30	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
162	161	eleq2d 2687	. . . . . . . . . . . 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 1403	. . . . . . . . . . 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 741	. . . . . . . . . 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 3152	. . . . . . . . 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 3152	. . . . . . . 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 3152	. . . . . . 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 3152	. . . . . 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 298	. . . . 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 1399	. . . 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 303	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
172	171	ex 450	. 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 369	1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 ._rcmulr 15942 Scalarcsca 15944 ·_𝑠 cvsca 15945 Grpcgrp 17422 1_rcur 18501 Ringcrg 18547 LModclmod 18863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865

This theorem is referenced by: lmodpropd 18926 lvecprop2d 19166

W3C validator