Theorem assapropd 22002

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 21991	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ LMod)
2		assaring 21992	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ Ring)
3	1, 2	jca 520	. . 3 ⊢ (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
5		assalmod 21991	. . . 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 21027	. . . 4 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
14	5, 13	imbitrrid 249	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ LMod))
15		assaring 21992	. . . 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 20372	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
18	15, 17	imbitrrid 249	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ Ring))
19	14, 18	jcad 521	. 2 ⊢ (𝜑 → (𝐿 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
20	13, 17	anbi12d 643	. . . . . 6 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
21	20	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
22		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑)
23		simplrl 788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ LMod)
24		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃)
25	9	fveq2d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾)))
26	11, 25	eqtrid 2810	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
27	22, 26	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑃 = (Base‘(Scalar‘𝐾)))
28	24, 27	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘(Scalar‘𝐾)))
29		simprrl 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ 𝐵)
30	22, 6	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐵 = (Base‘𝐾))
31	29, 30	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾))
32		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
33		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
34		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
35		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
36	32, 33, 34, 35	lmodvscl 20980	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
37	23, 28, 31, 36	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
38	37, 30	eleqtrrd 2866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ 𝐵)
39		simprrr 793	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵)
40	16	oveqrspc2v 7439	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑧) = (𝑟( ·𝑠 ‘𝐿)𝑧))
43	22, 24, 29, 42	syl12anc 849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) = (𝑟( ·𝑠 ‘𝐿)𝑧))
44	43	oveq1d 7427	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝐾) = (.r‘𝐾)
47		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Ring)
48	39, 30	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾))
49	32, 46, 47, 31, 48	ringcld 20343	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
50	49, 30	eleqtrrd 2866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ 𝐵)
51	12	oveqrspc2v 7439	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤))
54	22, 29, 39, 53	syl12anc 849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤))
55	54	oveq2d 7428	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 2779	. . . . . . . . . 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 20980	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
59	23, 28, 48, 58	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
60	59, 30	eleqtrrd 2866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ 𝐵)
61	16	oveqrspc2v 7439	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑤) = (𝑟( ·𝑠 ‘𝐿)𝑤))
64	22, 24, 39, 63	syl12anc 849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) = (𝑟( ·𝑠 ‘𝐿)𝑤))
65	64	oveq2d 7428	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 2779	. . . . . . . . . 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 643	. . . . . . . . 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 472	. . . . . . . 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 3227	. . . . . . 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 3186	. . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐾)))
73	6	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐾))
74	73	raleqdv 3323	. . . . . . . 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 3338	. . . . . . 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 3338	. . . . . 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 6887	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
78	11, 77	eqtrid 2810	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
79	78	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐿)))
80	7	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿))
81	80	raleqdv 3323	. . . . . . . 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 3338	. . . . . . 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 3338	. . . . . 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 312	. . . . 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 643	. . . 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 21987	. . . 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 2763	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
88		eqid 2763	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
89		eqid 2763	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
90		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
91		eqid 2763	. . . . 5 ⊢ (.r‘𝐿) = (.r‘𝐿)
92	87, 88, 89, 90, 91	isassa 21987	. . . 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 317	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
94	93	ex 417	. 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)))
95	4, 19, 94	pm5.21ndd 382	1 ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  ._rcmulr 17312  Scalarcsca 17314   ·_𝑠 cvsca 17315  Ringcrg 20316  LModclmod 20962  AssAlgcasa 21981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-plusg 17324  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-mgp 20218  df-ur 20265  df-ring 20318  df-lmod 20964  df-assa 21984

This theorem is referenced by:  opsrassa  22192