Theorem assapropd 15123

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 15115	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ LMod)
2		assaring 15116	. . . 4 ⊢ (𝐾 ∈ AssAlg → 𝐾 ∈ Ring)
3	1, 2	jca 306	. . 3 ⊢ (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring))
4	3	a1i 9	. 2 ⊢ (𝜑 → (𝐾 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
5		assalmod 15115	. . . 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 14795	. . . 4 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
14	5, 13	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ LMod))
15		assaring 15116	. . . 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 14450	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
18	15, 17	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝐿 ∈ AssAlg → 𝐾 ∈ Ring))
19	14, 18	jcad 307	. 2 ⊢ (𝜑 → (𝐿 ∈ AssAlg → (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)))
20	13, 17	anbi12d 477	. . . . . 6 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
21	20	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) ↔ (𝐿 ∈ LMod ∧ 𝐿 ∈ Ring)))
22		simpll 531	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝜑)
23		simplrl 541	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ LMod)
24		simprl 535	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ 𝑃)
25	9	fveq2d 5702	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾)))
26	11, 25	eqtrid 2283	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
27	22, 26	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑃 = (Base‘(Scalar‘𝐾)))
28	24, 27	eleqtrd 2317	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑟 ∈ (Base‘(Scalar‘𝐾)))
29		simprrl 545	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ 𝐵)
30	22, 6	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐵 = (Base‘𝐾))
31	29, 30	eleqtrd 2317	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑧 ∈ (Base‘𝐾))
32		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
33		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
34		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
35		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐾))
36	32, 33, 34, 35	lmodvscl 14750	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
37	23, 28, 31, 36	syl3anc 1278	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ (Base‘𝐾))
38	37, 30	eleqtrrd 2318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) ∈ 𝐵)
39		simprrr 546	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ 𝐵)
40	16	oveqrspc2v 6115	. . . . . . . . . . . . 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 1276	. . . . . . . . . . . 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 6115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑧 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑧) = (𝑟( ·𝑠 ‘𝐿)𝑧))
43	22, 24, 29, 42	syl12anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑧) = (𝑟( ·𝑠 ‘𝐿)𝑧))
44	43	oveq1d 6103	. . . . . . . . . . . 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 2271	. . . . . . . . . . 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 2238	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝐾) = (.r‘𝐾)
47		simplrr 542	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝐾 ∈ Ring)
48	39, 30	eleqtrd 2317	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → 𝑤 ∈ (Base‘𝐾))
49	32, 46, 47, 31, 48	ringcld 14429	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
50	49, 30	eleqtrrd 2318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) ∈ 𝐵)
51	12	oveqrspc2v 6115	. . . . . . . . . . . . 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 1276	. . . . . . . . . . . 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 6115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤))
54	22, 29, 39, 53	syl12anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑧(.r‘𝐾)𝑤) = (𝑧(.r‘𝐿)𝑤))
55	54	oveq2d 6104	. . . . . . . . . . . 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 2271	. . . . . . . . . . 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 2253	. . . . . . . . . 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 14750	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
59	23, 28, 48, 58	syl3anc 1278	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ (Base‘𝐾))
60	59, 30	eleqtrrd 2318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) ∈ 𝐵)
61	16	oveqrspc2v 6115	. . . . . . . . . . . . 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 1276	. . . . . . . . . . . 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 6115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑃 ∧ 𝑤 ∈ 𝐵)) → (𝑟( ·𝑠 ‘𝐾)𝑤) = (𝑟( ·𝑠 ‘𝐿)𝑤))
64	22, 24, 39, 63	syl12anc 1276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) ∧ (𝑟 ∈ 𝑃 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) → (𝑟( ·𝑠 ‘𝐾)𝑤) = (𝑟( ·𝑠 ‘𝐿)𝑤))
65	64	oveq2d 6104	. . . . . . . . . . . 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 2271	. . . . . . . . . . 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 2253	. . . . . . . . . 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 477	. . . . . . . . 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 404	. . . . . . . 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 2572	. . . . . . 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 2546	. . . . . 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 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐾)))
73	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐾))
74	73	raleqdv 2755	. . . . . . . 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 2765	. . . . . . 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 2765	. . . . . 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 5702	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
78	11, 77	eqtrid 2283	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
79	78	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝑃 = (Base‘(Scalar‘𝐿)))
80	7	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → 𝐵 = (Base‘𝐿))
81	80	raleqdv 2755	. . . . . . . 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 2765	. . . . . . 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 2765	. . . . . 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 218	. . . . 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 477	. . . 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 15111	. . . 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 2238	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
88		eqid 2238	. . . . 5 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
89		eqid 2238	. . . . 5 ⊢ (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
90		eqid 2238	. . . . 5 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
91		eqid 2238	. . . . 5 ⊢ (.r‘𝐿) = (.r‘𝐿)
92	87, 88, 89, 90, 91	isassa 15111	. . . 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 223	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ LMod ∧ 𝐾 ∈ Ring)) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
94	93	ex 115	. 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐾 ∈ Ring) → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg)))
95	4, 19, 94	pm5.21ndd 717	1 ⊢ (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  +_gcplusg 13490  ._rcmulr 13491  Scalarcsca 13493   ·_𝑠 cvsca 13494  Ringcrg 14407  LModclmod 14732  AssAlgcasa 15105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-sca 13506  df-vsca 13507  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-mgp 14325  df-ur 14370  df-ring 14409  df-lmod 14734  df-assa 15108

This theorem is used by: (None)