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Theorem assapropd 21766

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 21755	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ LMod)
2		assaring 21756	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ Ring)
3	1, 2	jca 511	. . 3 ⊢ (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
5		assalmod 21755	. . . 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 20771	. . . 4 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
14	5, 13	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ LMod))
15		assaring 21756	. . . 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 20187	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
18	15, 17	imbitrrid 245	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ Ring))
19	14, 18	jcad 512	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
22		simpll 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑)
23		simplrl 774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ LMod)
24		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃)
25	9	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾)))
26	11, 25	eqtrid 2778	. . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘(Scalar‘𝐾)))
29		simprrl 778	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾))
32		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
33		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
34		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘𝐾) = ( ·_𝑠 ‘𝐾)
35		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
36	32, 33, 34, 35	lmodvscl 20724	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) ∈ 𝐵)
39		simprrr 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵)
40	16	oveqrspc2v 7432	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
43	22, 24, 29, 42	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑧) = (𝑟( ·_𝑠 ‘𝐿)𝑧))
44	43	oveq1d 7420	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 2726	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐾) = (._r‘𝐾)
47		simplrr 775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Ring)
48	39, 30	eleqtrd 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾))
49	32, 46, 47, 31, 48	ringcld 20162	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
50	49, 30	eleqtrrd 2830	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) ∈ 𝐵)
51	12	oveqrspc2v 7432	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(._r‘𝐾)𝑤) = (𝑧(._r‘𝐿)𝑤))
54	22, 29, 39, 53	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(._r‘𝐾)𝑤) = (𝑧(._r‘𝐿)𝑤))
55	54	oveq2d 7421	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 2742	. . . . . . . . . 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 20724	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) ∈ 𝐵)
61	16	oveqrspc2v 7432	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
64	22, 24, 39, 63	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·_𝑠 ‘𝐾)𝑤) = (𝑟( ·_𝑠 ‘𝐿)𝑤))
65	64	oveq2d 7421	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 2742	. . . . . . . . . 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 467	. . . . . . . 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 3210	. . . . . . 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 3169	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐾)))
73	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐾))
74	73	raleqdv 3319	. . . . . . . 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 3336	. . . . . . 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 3336	. . . . . 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 6889	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
78	11, 77	eqtrid 2778	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐿)))
80	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿))
81	80	raleqdv 3319	. . . . . . . 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 3336	. . . . . . 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 3336	. . . . . 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 309	. . . . 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 21751	. . . 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 2726	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
88		eqid 2726	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
89		eqid 2726	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
90		eqid 2726	. . . . 5 ⊢ ( ·_𝑠 ‘𝐿) = ( ·_𝑠 ‘𝐿)
91		eqid 2726	. . . . 5 ⊢ (._r‘𝐿) = (._r‘𝐿)
92	87, 88, 89, 90, 91	isassa 21751	. . . 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 314	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
94	93	ex 412	. 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)))
95	4, 19, 94	pm5.21ndd 379	1 ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +_gcplusg 17206 ._rcmulr 17207 Scalarcsca 17209 ·_𝑠 cvsca 17210 Ringcrg 20138 LModclmod 20706 AssAlgcasa 21745

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 df-assa 21748

This theorem is referenced by: opsrassa 21963

W3C validator