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Theorem assapropd 21809

Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)

**Hypotheses**
Ref	Expression
assapropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
assapropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
assapropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
assapropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦))
assapropd.5	⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))
assapropd.6	⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
assapropd.7	⊢ 𝑃 = (Base‘𝐹)
assapropd.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))

**Assertion**
Ref	Expression
assapropd	⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))

Distinct variable groups: 𝑥,𝑦,𝐾 𝑥,𝐿,𝑦 𝑥,𝑃,𝑦 𝜑,𝑥,𝑦 𝑥,𝐵,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

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

Step	Hyp	Ref	Expression
1		assalmod 21798	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ LMod)
2		assaring 21799	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ Ring)
3	1, 2	jca 510	. . 3 ⊢ (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
5		assalmod 21798	. . . 4 ⊢ (𝐿 ∈ AssAlg → 𝐿 ∈ LMod)
6		assapropd.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
7		assapropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
8		assapropd.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
9		assapropd.5	. . . . 5 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))
10		assapropd.6	. . . . 5 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
11		assapropd.7	. . . . 5 ⊢ 𝑃 = (Base‘𝐹)
12		assapropd.8	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·_𝑠 ‘𝐾)𝑦) = (𝑥( ·_𝑠 ‘𝐿)𝑦))
13	6, 7, 8, 9, 10, 11, 12	lmodpropd 20812	. . . 4 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
14	5, 13	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ LMod))
15		assaring 21799	. . . 4 ⊢ (𝐿 ∈ AssAlg → 𝐿 ∈ Ring)
16		assapropd.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦))
17	6, 7, 8, 16	ringpropd 20228	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
18	15, 17	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ Ring))
19	14, 18	jcad 511	. 2 ⊢ (𝜑 → (𝐿 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
20	13, 17	anbi12d 630	. . . . . 6 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
21	20	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
22		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑)
23		simplrl 775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ LMod)
24		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃)
25	9	fveq2d 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾)))
26	11, 25	eqtrid 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
27	22, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑃 = (Base‘(Scalar‘𝐾)))
28	24, 27	eleqtrd 2827	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘(Scalar‘𝐾)))
29		simprrl 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ 𝐵)
30	22, 6	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐵 = (Base‘𝐾))
31	29, 30	eleqtrd 2827	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾))
32		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
33		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
34		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
35		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
36	32, 33, 34, 35	lmodvscl 20765	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
37	23, 28, 31, 36	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
38	37, 30	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵)
39		simprrr 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵)
40	16	oveqrspc2v 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐿)𝑤))
41	22, 38, 39, 40	syl12anc 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐿)𝑤))
42	12	oveqrspc2v 7443	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
43	22, 24, 29, 42	syl12anc 835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
44	43	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐿)𝑤) = ((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤))
45	41, 44	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = ((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤))
46		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐾) = (._r‘𝐾)
47		simplrr 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Ring)
48	39, 30	eleqtrd 2827	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾))
49	32, 46, 47, 31, 48	ringcld 20203	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
50	49, 30	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) ∈ 𝐵)
51	12	oveqrspc2v 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ (𝑧(._r‘𝐾)𝑤) ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐾)𝑤)))
52	22, 24, 50, 51	syl12anc 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐾)𝑤)))
53	16	oveqrspc2v 7443	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(._r‘𝐾)𝑤) = (𝑧(._r‘𝐿)𝑤))
54	22, 29, 39, 53	syl12anc 835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) = (𝑧(._r‘𝐿)𝑤))
55	54	oveq2d 7432	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))
56	52, 55	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))
57	45, 56	eqeq12d 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ↔ ((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))))
58	32, 33, 34, 35	lmodvscl 20765	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
59	23, 28, 48, 58	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
60	59, 30	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
61	16	oveqrspc2v 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)) → (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
62	22, 29, 60, 61	syl12anc 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)))
63	12	oveqrspc2v 7443	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
64	22, 24, 39, 63	syl12anc 835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
65	64	oveq2d 7432	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
66	62, 65	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)))
67	66, 56	eqeq12d 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ↔ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))))
68	57, 67	anbi12d 630	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → ((((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
69	68	anassrs 466	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ 𝑟 ∈ 𝑃) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
70	69	2ralbidva 3207	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ 𝑟 ∈ 𝑃) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
71	70	ralbidva 3166	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
72	26	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐾)))
73	6	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐾))
74	73	raleqdv 3315	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)))))
75	73, 74	raleqbidv 3330	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)))))
76	72, 75	raleqbidv 3330	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)))))
77	10	fveq2d 6896	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
78	11, 77	eqtrid 2777	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
79	78	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐿)))
80	7	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿))
81	80	raleqdv 3315	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
82	80, 81	raleqbidv 3330	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))) ↔ ∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
83	79, 82	raleqbidv 3330	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ 𝑃 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
84	71, 76, 83	3bitr3d 308	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (∀𝑟 ∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
85	21, 84	anbi12d 630	. . . 4 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)))) ↔ ((𝐿 ∈ LMod ∧ 𝐿 ∈ Ring) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤))))))
86	32, 33, 35, 34, 46	isassa 21794	. . . 4 ⊢ (𝐾 ∈ AssAlg ↔ ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝐾))∀𝑧 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)(((𝑟( ·_𝑠 ‘𝐾)𝑧)(._r‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)) ∧ (𝑧(._r‘𝐾)(𝑟( ·_𝑠 ‘𝐾)𝑤)) = (𝑟( ·_𝑠 ‘𝐾)(𝑧(._r‘𝐾)𝑤)))))
87		eqid 2725	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
88		eqid 2725	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
89		eqid 2725	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
90		eqid 2725	. . . . 5 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
91		eqid 2725	. . . . 5 ⊢ (._r‘𝐿) = (._r‘𝐿)
92	87, 88, 89, 90, 91	isassa 21794	. . . 4 ⊢ (𝐿 ∈ AssAlg ↔ ((𝐿 ∈ LMod ∧ 𝐿 ∈ Ring) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝐿))∀𝑧 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)(((𝑟( ·_𝑠 ‘𝐿)𝑧)(._r‘𝐿)𝑤) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)) ∧ (𝑧(._r‘𝐿)(𝑟( ·_𝑠 ‘𝐿)𝑤)) = (𝑟( ·_𝑠 ‘𝐿)(𝑧(._r‘𝐿)𝑤)))))
93	85, 86, 92	3bitr4g 313	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
94	93	ex 411	. 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)))
95	4, 19, 94	pm5.21ndd 378	1 ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 Scalarcsca 17235 ·_𝑠 cvsca 17236 Ringcrg 20177 LModclmod 20747 AssAlgcasa 21788

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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-assa 21791

This theorem is referenced by: opsrassa 22011

W3C validator