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Theorem lmodprop2d 20770

Description: If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 20771 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 20713	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐾 ∈ Grp)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐾 ∈ Grp))
3		eqid 2726	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2726	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
5		eqid 2726	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
6		lmodprop2d.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝐾)
7		eqid 2726	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
8		eqid 2726	. . . . . 6 ⊢ (+_g‘𝐹) = (+_g‘𝐹)
9		eqid 2726	. . . . . 6 ⊢ (._r‘𝐹) = (._r‘𝐹)
10		eqid 2726	. . . . . 6 ⊢ (1_r‘𝐹) = (1_r‘𝐹)
11	3, 4, 5, 6, 7, 8, 9, 10	islmod 20710	. . . . 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 1143	. . . 4 ⊢ (𝐾 ∈ LMod → 𝐹 ∈ Ring)
13	12	a1i 11	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → 𝐹 ∈ Ring))
14		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ LMod)
15		simprl 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
16		lmodprop2d.p1	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
17	16	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
18	15, 17	eleqtrd 2829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐹))
19		simprr 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
20		lmodprop2d.b1	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
21	20	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
22	19, 21	eleqtrd 2829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
23	3, 6, 5, 7	lmodvscl 20724	. . . . . . 7 ⊢ ((𝐾 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
24	14, 18, 22, 23	syl3anc 1368	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾))
25	24, 21	eleqtrrd 2830	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
26	25	ralrimivva 3194	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
27	26	ex 412	. . 3 ⊢ (𝜑 → (𝐾 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
28	2, 13, 27	3jcad 1126	. 2 ⊢ (𝜑 → (𝐾 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
29		lmodgrp 20713	. . . 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 18881	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
33	29, 32	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐾 ∈ Grp))
34		eqid 2726	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
35		eqid 2726	. . . . . 6 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
36		eqid 2726	. . . . . 6 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
37		lmodprop2d.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝐿)
38		eqid 2726	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
39		eqid 2726	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
40		eqid 2726	. . . . . 6 ⊢ (._r‘𝐺) = (._r‘𝐺)
41		eqid 2726	. . . . . 6 ⊢ (1_r‘𝐺) = (1_r‘𝐺)
42	34, 35, 36, 37, 38, 39, 40, 41	islmod 20710	. . . . 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 1143	. . . 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 20187	. . . 4 ⊢ (𝜑 → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
48	43, 47	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → 𝐹 ∈ Ring))
49		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ LMod)
50		simprl 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑃)
51	44	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
52	50, 51	eleqtrd 2829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐺))
53		simprr 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
54	30	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
55	53, 54	eleqtrd 2829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
56	34, 37, 36, 38	lmodvscl 20724	. . . . . . 7 ⊢ ((𝐿 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥( ·_𝑠 ‘𝐿)𝑦) ∈ (Base‘𝐿))
57	49, 52, 55, 56	syl3anc 1368	. . . . . 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 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
60	57, 59, 54	3eltr4d 2842	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ LMod) ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
61	60	ralrimivva 3194	. . . 4 ⊢ ((𝜑 ∧ 𝐿 ∈ LMod) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)
62	61	ex 412	. . 3 ⊢ (𝜑 → (𝐿 ∈ LMod → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵))
63	33, 48, 62	3jcad 1126	. 2 ⊢ (𝜑 → (𝐿 ∈ LMod → (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)))
64	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
65	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐹 ∈ Ring ↔ 𝐺 ∈ Ring))
66		simpll 764	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
70	66, 67, 68, 69	syl12anc 834	. . . . . . . . . . . 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 2812	. . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 18871	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑤 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑤(+_g‘𝐾)𝑧) ∈ (Base‘𝐾))
78	72, 74, 76, 77	syl3anc 1368	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑤(+_g‘𝐾)𝑧) = (𝑤(+_g‘𝐿)𝑧))
83	66, 68, 75, 82	syl12anc 834	. . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
94	66, 67, 75, 93	syl12anc 834	. . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . 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 20177	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(+_g‘𝐹)𝑟) ∈ (Base‘𝐹))
104	98, 101, 102, 103	syl3anc 1368	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(+_g‘𝐹)𝑟) = (𝑞(+_g‘𝐺)𝑟))
109	108	ad2ant2r 744	. . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑞( ·_𝑠 ‘𝐾)𝑤) = (𝑞( ·_𝑠 ‘𝐿)𝑤))
117	66, 99, 68, 116	syl12anc 834	. . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) ∧ ((𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑞( ·_𝑠 ‘𝐾)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = ((𝑞( ·_𝑠 ‘𝐿)𝑤)(+_g‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
119	115, 118	eqtrd 2766	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 1432	. . . . . . . . . 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 20155	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ (Base‘𝐹) ∧ 𝑟 ∈ (Base‘𝐹)) → (𝑞(._r‘𝐹)𝑟) ∈ (Base‘𝐹))
123	98, 101, 102, 122	syl3anc 1368	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (𝑞(._r‘𝐹)𝑟) = (𝑞(._r‘𝐺)𝑟))
128	127	ad2ant2r 744	. . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 20165	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . . 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 20317	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝐹) = (1_r‘𝐺))
142	141	ad2antrr 723	. . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 2728	. . . . . . . . . . 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 630	. . . . . . . . . 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 630	. . . . . . . . 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 467	. . . . . . . 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 3210	. . . . . . 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 3210	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐹))
152	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
153	152	eleq2d 2813	. . . . . . . . . . . 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 1436	. . . . . . . . . . 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 629	. . . . . . . . . 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 3336	. . . . . . . . 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 3336	. . . . . . . 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 3336	. . . . . . 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 3336	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝑃 = (Base‘𝐺))
161	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
162	161	eleq2d 2813	. . . . . . . . . . . 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 1436	. . . . . . . . . . 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 629	. . . . . . . . . 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 3336	. . . . . . . . 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 3336	. . . . . . . 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 3336	. . . . . . 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 3336	. . . . . 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 309	. . . . 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 1432	. . . 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 314	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝐵 (𝑥( ·_𝑠 ‘𝐾)𝑦) ∈ 𝐵)) → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
172	171	ex 412	. 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 379	1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +_gcplusg 17206 ._rcmulr 17207 Scalarcsca 17209 ·_𝑠 cvsca 17210 Grpcgrp 18863 1_rcur 20086 Ringcrg 20138 LModclmod 20706

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708

This theorem is referenced by: lmodpropd 20771 lvecprop2d 21017

W3C validator