Theorem dvhlveclem 39122

Description: Lemma for dvhlvec 39123. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does 𝜑 → method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
dvhgrp.b	⊢ 𝐵 = (Base‘𝐾)
dvhgrp.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhgrp.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhgrp.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhgrp.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhgrp.d	⊢ 𝐷 = (Scalar‘𝑈)
dvhgrp.p	⊢ ⨣ = (+g‘𝐷)
dvhgrp.a	⊢ + = (+g‘𝑈)
dvhgrp.o	⊢ 0 = (0g‘𝐷)
dvhgrp.i	⊢ 𝐼 = (invg‘𝐷)
dvhlvec.m	⊢ × = (.r‘𝐷)
dvhlvec.s	⊢ · = ( ·𝑠 ‘𝑈)

Assertion

Ref	Expression
dvhlveclem	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec)

Proof of Theorem dvhlveclem

Dummy variables 𝑡 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhgrp.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhgrp.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhgrp.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		dvhgrp.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2738	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 39101	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2744	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑇 × 𝐸) = (Base‘𝑈))
8		dvhgrp.a	. . . 4 ⊢ + = (+g‘𝑈)
9	8	a1i 11	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = (+g‘𝑈))
10		dvhgrp.d	. . . 4 ⊢ 𝐷 = (Scalar‘𝑈)
11	10	a1i 11	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = (Scalar‘𝑈))
12		dvhlvec.s	. . . 4 ⊢ · = ( ·𝑠 ‘𝑈)
13	12	a1i 11	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → · = ( ·𝑠 ‘𝑈))
14		eqid 2738	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	1, 3, 4, 10, 14	dvhbase 39097	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
16	15	eqcomd 2744	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷))
17		dvhgrp.p	. . . 4 ⊢ ⨣ = (+g‘𝐷)
18	17	a1i 11	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (+g‘𝐷))
19		dvhlvec.m	. . . 4 ⊢ × = (.r‘𝐷)
20	19	a1i 11	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → × = (.r‘𝐷))
21		eqid 2738	. . . . . 6 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
22	1, 21, 4, 10	dvhsca 39096	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
23	22	fveq2d 6778	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24		eqid 2738	. . . . 5 ⊢ (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
25	1, 2, 21, 24	erng1r 39009	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
26	23, 25	eqtr2d 2779	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) = (1r‘𝐷))
27	1, 21	erngdv 39007	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
28	22, 27	eqeltrd 2839	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
29		drngring 19998	. . . 4 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Ring)
30	28, 29	syl 17	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring)
31		dvhgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
32		dvhgrp.o	. . . 4 ⊢ 0 = (0g‘𝐷)
33		dvhgrp.i	. . . 4 ⊢ 𝐼 = (invg‘𝐷)
34	31, 1, 2, 3, 4, 10, 17, 8, 32, 33	dvhgrp 39121	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)
35	1, 2, 3, 4, 12	dvhvscacl 39117	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
36	35	3impb 1114	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸)) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
37		simpl 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38		simpr1 1193	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ 𝐸)
39		simpr2 1194	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (𝑇 × 𝐸))
40		xp1st 7863	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑇 × 𝐸) → (1st ‘𝑡) ∈ 𝑇)
41	39, 40	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘𝑡) ∈ 𝑇)
42		simpr3 1195	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
43		xp1st 7863	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1st ‘𝑓) ∈ 𝑇)
44	42, 43	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘𝑓) ∈ 𝑇)
45	1, 2, 3	tendospdi1 39034	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (1st ‘𝑡) ∈ 𝑇 ∧ (1st ‘𝑓) ∈ 𝑇)) → (𝑠‘((1st ‘𝑡) ∘ (1st ‘𝑓))) = ((𝑠‘(1st ‘𝑡)) ∘ (𝑠‘(1st ‘𝑓))))
46	37, 38, 41, 44, 45	syl13anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘((1st ‘𝑡) ∘ (1st ‘𝑓))) = ((𝑠‘(1st ‘𝑡)) ∘ (𝑠‘(1st ‘𝑓))))
47	1, 2, 3, 4, 10, 8, 17	dvhvadd 39106	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩)
48	47	3adantr1 1168	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩)
49	48	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = (1st ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩))
50		fvex 6787	. . . . . . . . . 10 ⊢ (1st ‘𝑡) ∈ V
51		fvex 6787	. . . . . . . . . 10 ⊢ (1st ‘𝑓) ∈ V
52	50, 51	coex 7777	. . . . . . . . 9 ⊢ ((1st ‘𝑡) ∘ (1st ‘𝑓)) ∈ V
53		ovex 7308	. . . . . . . . 9 ⊢ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ V
54	52, 53	op1st 7839	. . . . . . . 8 ⊢ (1st ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩) = ((1st ‘𝑡) ∘ (1st ‘𝑓))
55	49, 54	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st ‘𝑡) ∘ (1st ‘𝑓)))
56	55	fveq2d 6778	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = (𝑠‘((1st ‘𝑡) ∘ (1st ‘𝑓))))
57	1, 2, 3, 4, 12	dvhvsca 39115	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩)
58	57	3adantr3 1170	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩)
59	58	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (1st ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩))
60		fvex 6787	. . . . . . . . 9 ⊢ (𝑠‘(1st ‘𝑡)) ∈ V
61		vex 3436	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
62		fvex 6787	. . . . . . . . . 10 ⊢ (2nd ‘𝑡) ∈ V
63	61, 62	coex 7777	. . . . . . . . 9 ⊢ (𝑠 ∘ (2nd ‘𝑡)) ∈ V
64	60, 63	op1st 7839	. . . . . . . 8 ⊢ (1st ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩) = (𝑠‘(1st ‘𝑡))
65	59, 64	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (𝑠‘(1st ‘𝑡)))
66	1, 2, 3, 4, 12	dvhvsca 39115	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
67	66	3adantr2 1169	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
68	67	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩))
69		fvex 6787	. . . . . . . . 9 ⊢ (𝑠‘(1st ‘𝑓)) ∈ V
70		fvex 6787	. . . . . . . . . 10 ⊢ (2nd ‘𝑓) ∈ V
71	61, 70	coex 7777	. . . . . . . . 9 ⊢ (𝑠 ∘ (2nd ‘𝑓)) ∈ V
72	69, 71	op1st 7839	. . . . . . . 8 ⊢ (1st ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩) = (𝑠‘(1st ‘𝑓))
73	68, 72	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st ‘𝑓)))
74	65, 73	coeq12d 5773	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))) = ((𝑠‘(1st ‘𝑡)) ∘ (𝑠‘(1st ‘𝑓))))
75	46, 56, 74	3eqtr4d 2788	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))))
76	30	adantr 481	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
77	16	adantr 481	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
78	38, 77	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79		xp2nd 7864	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑇 × 𝐸) → (2nd ‘𝑡) ∈ 𝐸)
80	39, 79	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑡) ∈ 𝐸)
81	80, 77	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑡) ∈ (Base‘𝐷))
82		xp2nd 7864	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2nd ‘𝑓) ∈ 𝐸)
83	42, 82	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ 𝐸)
84	83, 77	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ (Base‘𝐷))
85	14, 17, 19	ringdi 19805	. . . . . . . 8 ⊢ ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ (2nd ‘𝑡) ∈ (Base‘𝐷) ∧ (2nd ‘𝑓) ∈ (Base‘𝐷))) → (𝑠 × ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))) = ((𝑠 × (2nd ‘𝑡)) ⨣ (𝑠 × (2nd ‘𝑓))))
86	76, 78, 81, 84, 85	syl13anc 1371	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))) = ((𝑠 × (2nd ‘𝑡)) ⨣ (𝑠 × (2nd ‘𝑓))))
87	14, 17	ringacl 19817	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Ring ∧ (2nd ‘𝑡) ∈ (Base‘𝐷) ∧ (2nd ‘𝑓) ∈ (Base‘𝐷)) → ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ (Base‘𝐷))
88	76, 81, 84, 87	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ (Base‘𝐷))
89	88, 77	eleqtrrd 2842	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ 𝐸)
90	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ 𝐸)) → (𝑠 × ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))) = (𝑠 ∘ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))))
91	37, 38, 89, 90	syl12anc 834	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))) = (𝑠 ∘ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))))
92	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd ‘𝑡) ∈ 𝐸)) → (𝑠 × (2nd ‘𝑡)) = (𝑠 ∘ (2nd ‘𝑡)))
93	37, 38, 80, 92	syl12anc 834	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd ‘𝑡)) = (𝑠 ∘ (2nd ‘𝑡)))
94	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (𝑠 × (2nd ‘𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
95	37, 38, 83, 94	syl12anc 834	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd ‘𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
96	93, 95	oveq12d 7293	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd ‘𝑡)) ⨣ (𝑠 × (2nd ‘𝑓))) = ((𝑠 ∘ (2nd ‘𝑡)) ⨣ (𝑠 ∘ (2nd ‘𝑓))))
97	86, 91, 96	3eqtr3d 2786	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))) = ((𝑠 ∘ (2nd ‘𝑡)) ⨣ (𝑠 ∘ (2nd ‘𝑓))))
98	48	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = (2nd ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩))
99	52, 53	op2nd 7840	. . . . . . . 8 ⊢ (2nd ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩) = ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))
100	98, 99	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)))
101	100	coeq2d 5771	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = (𝑠 ∘ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))))
102	58	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (2nd ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩))
103	60, 63	op2nd 7840	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩) = (𝑠 ∘ (2nd ‘𝑡))
104	102, 103	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd ‘𝑡)))
105	67	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩))
106	69, 71	op2nd 7840	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩) = (𝑠 ∘ (2nd ‘𝑓))
107	105, 106	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
108	104, 107	oveq12d 7293	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑡)) ⨣ (2nd ‘(𝑠 · 𝑓))) = ((𝑠 ∘ (2nd ‘𝑡)) ⨣ (𝑠 ∘ (2nd ‘𝑓))))
109	97, 101, 108	3eqtr4d 2788	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = ((2nd ‘(𝑠 · 𝑡)) ⨣ (2nd ‘(𝑠 · 𝑓))))
110	75, 109	opeq12d 4812	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩ = ⟨((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))), ((2nd ‘(𝑠 · 𝑡)) ⨣ (2nd ‘(𝑠 · 𝑓)))⟩)
111	1, 2, 3, 4, 10, 17, 8	dvhvaddcl 39109	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
112	111	3adantr1 1168	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
113	1, 2, 3, 4, 12	dvhvsca 39115	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
114	37, 38, 112, 113	syl12anc 834	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
115	35	3adantr3 1170	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
116	1, 2, 3, 4, 12	dvhvscacl 39117	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
117	116	3adantr2 1169	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
118	1, 2, 3, 4, 10, 8, 17	dvhvadd 39106	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠 · 𝑡) ∈ (𝑇 × 𝐸) ∧ (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑡) + (𝑠 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))), ((2nd ‘(𝑠 · 𝑡)) ⨣ (2nd ‘(𝑠 · 𝑓)))⟩)
119	37, 115, 117, 118	syl12anc 834	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑡) + (𝑠 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))), ((2nd ‘(𝑠 · 𝑡)) ⨣ (2nd ‘(𝑠 · 𝑓)))⟩)
120	110, 114, 119	3eqtr4d 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121		simpl 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
122		simpr1 1193	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ 𝐸)
123		simpr2 1194	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ 𝐸)
124		simpr3 1195	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
125	124, 43	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘𝑓) ∈ 𝑇)
126		eqid 2738	. . . . . . . 8 ⊢ (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
127	1, 2, 3, 21, 126	erngplus2 38818	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ (1st ‘𝑓) ∈ 𝑇)) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st ‘𝑓)) = ((𝑠‘(1st ‘𝑓)) ∘ (𝑡‘(1st ‘𝑓))))
128	121, 122, 123, 125, 127	syl13anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st ‘𝑓)) = ((𝑠‘(1st ‘𝑓)) ∘ (𝑡‘(1st ‘𝑓))))
129	22	fveq2d 6778	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
130	17, 129	eqtrid 2790	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (+g‘((EDRing‘𝐾)‘𝑊)))
131	130	oveqd 7292	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠 ⨣ 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132	131	fveq1d 6776	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st ‘𝑓)))
133	132	adantr 481	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st ‘𝑓)))
134	66	3adantr2 1169	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
135	134	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩))
136	135, 72	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st ‘𝑓)))
137	1, 2, 3, 4, 12	dvhvsca 39115	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
138	137	3adantr1 1168	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
139	138	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩))
140		fvex 6787	. . . . . . . . 9 ⊢ (𝑡‘(1st ‘𝑓)) ∈ V
141		vex 3436	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
142	141, 70	coex 7777	. . . . . . . . 9 ⊢ (𝑡 ∘ (2nd ‘𝑓)) ∈ V
143	140, 142	op1st 7839	. . . . . . . 8 ⊢ (1st ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = (𝑡‘(1st ‘𝑓))
144	139, 143	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st ‘𝑓)))
145	136, 144	coeq12d 5773	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))) = ((𝑠‘(1st ‘𝑓)) ∘ (𝑡‘(1st ‘𝑓))))
146	128, 133, 145	3eqtr4d 2788	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)) = ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))))
147	30	adantr 481	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
148	16	adantr 481	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
149	122, 148	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150	123, 148	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (Base‘𝐷))
151	124, 82	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ 𝐸)
152	151, 148	eleqtrd 2841	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ (Base‘𝐷))
153	14, 17, 19	ringdir 19806	. . . . . . . 8 ⊢ ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷) ∧ (2nd ‘𝑓) ∈ (Base‘𝐷))) → ((𝑠 ⨣ 𝑡) × (2nd ‘𝑓)) = ((𝑠 × (2nd ‘𝑓)) ⨣ (𝑡 × (2nd ‘𝑓))))
154	147, 149, 150, 152, 153	syl13anc 1371	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) × (2nd ‘𝑓)) = ((𝑠 × (2nd ‘𝑓)) ⨣ (𝑡 × (2nd ‘𝑓))))
155	14, 17	ringacl 19817	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 ⨣ 𝑡) ∈ (Base‘𝐷))
156	147, 149, 150, 155	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ⨣ 𝑡) ∈ (Base‘𝐷))
157	156, 148	eleqtrrd 2842	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ⨣ 𝑡) ∈ 𝐸)
158	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠 ⨣ 𝑡) ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸)) → ((𝑠 ⨣ 𝑡) × (2nd ‘𝑓)) = ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓)))
159	121, 157, 151, 158	syl12anc 834	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) × (2nd ‘𝑓)) = ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓)))
160	121, 122, 151, 94	syl12anc 834	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd ‘𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
161	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (𝑡 × (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
162	121, 123, 151, 161	syl12anc 834	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
163	160, 162	oveq12d 7293	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd ‘𝑓)) ⨣ (𝑡 × (2nd ‘𝑓))) = ((𝑠 ∘ (2nd ‘𝑓)) ⨣ (𝑡 ∘ (2nd ‘𝑓))))
164	154, 159, 163	3eqtr3d 2786	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓)) = ((𝑠 ∘ (2nd ‘𝑓)) ⨣ (𝑡 ∘ (2nd ‘𝑓))))
165	134	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩))
166	165, 106	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
167	138	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (2nd ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩))
168	140, 142	op2nd 7840	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = (𝑡 ∘ (2nd ‘𝑓))
169	167, 168	eqtrdi 2794	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
170	166, 169	oveq12d 7293	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑓)) ⨣ (2nd ‘(𝑡 · 𝑓))) = ((𝑠 ∘ (2nd ‘𝑓)) ⨣ (𝑡 ∘ (2nd ‘𝑓))))
171	164, 170	eqtr4d 2781	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓)) = ((2nd ‘(𝑠 · 𝑓)) ⨣ (2nd ‘(𝑡 · 𝑓))))
172	146, 171	opeq12d 4812	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)), ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓))⟩ = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) ⨣ (2nd ‘(𝑡 · 𝑓)))⟩)
173	1, 2, 3, 4, 12	dvhvsca 39115	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠 ⨣ 𝑡) ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) · 𝑓) = ⟨((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)), ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓))⟩)
174	121, 157, 124, 173	syl12anc 834	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) · 𝑓) = ⟨((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)), ((𝑠 ⨣ 𝑡) ∘ (2nd ‘𝑓))⟩)
175	116	3adantr2 1169	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
176	1, 2, 3, 4, 12	dvhvscacl 39117	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
177	176	3adantr1 1168	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
178	1, 2, 3, 4, 10, 8, 17	dvhvadd 39106	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠 · 𝑓) ∈ (𝑇 × 𝐸) ∧ (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑓) + (𝑡 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) ⨣ (2nd ‘(𝑡 · 𝑓)))⟩)
179	121, 175, 177, 178	syl12anc 834	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑓) + (𝑡 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) ⨣ (2nd ‘(𝑡 · 𝑓)))⟩)
180	172, 174, 179	3eqtr4d 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
181	1, 2, 3	tendocoval 38780	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) ∧ (1st ‘𝑓) ∈ 𝑇) → ((𝑠 ∘ 𝑡)‘(1st ‘𝑓)) = (𝑠‘(𝑡‘(1st ‘𝑓))))
182	121, 122, 123, 125, 181	syl121anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡)‘(1st ‘𝑓)) = (𝑠‘(𝑡‘(1st ‘𝑓))))
183		coass 6169	. . . . . . 7 ⊢ ((𝑠 ∘ 𝑡) ∘ (2nd ‘𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd ‘𝑓)))
184	183	a1i 11	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡) ∘ (2nd ‘𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd ‘𝑓))))
185	182, 184	opeq12d 4812	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠 ∘ 𝑡)‘(1st ‘𝑓)), ((𝑠 ∘ 𝑡) ∘ (2nd ‘𝑓))⟩ = ⟨(𝑠‘(𝑡‘(1st ‘𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd ‘𝑓)))⟩)
186	1, 3	tendococl 38786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠 ∘ 𝑡) ∈ 𝐸)
187	121, 122, 123, 186	syl3anc 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ 𝑡) ∈ 𝐸)
188	1, 2, 3, 4, 12	dvhvsca 39115	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠 ∘ 𝑡) ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡) · 𝑓) = ⟨((𝑠 ∘ 𝑡)‘(1st ‘𝑓)), ((𝑠 ∘ 𝑡) ∘ (2nd ‘𝑓))⟩)
189	121, 187, 124, 188	syl12anc 834	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡) · 𝑓) = ⟨((𝑠 ∘ 𝑡)‘(1st ‘𝑓)), ((𝑠 ∘ 𝑡) ∘ (2nd ‘𝑓))⟩)
190	1, 2, 3	tendocl 38781	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ (1st ‘𝑓) ∈ 𝑇) → (𝑡‘(1st ‘𝑓)) ∈ 𝑇)
191	121, 123, 125, 190	syl3anc 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st ‘𝑓)) ∈ 𝑇)
192	1, 3	tendococl 38786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd ‘𝑓)) ∈ 𝐸)
193	121, 123, 151, 192	syl3anc 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd ‘𝑓)) ∈ 𝐸)
194	1, 2, 3, 4, 12	dvhopvsca 39116	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡‘(1st ‘𝑓)) ∈ 𝑇 ∧ (𝑡 ∘ (2nd ‘𝑓)) ∈ 𝐸)) → (𝑠 · ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st ‘𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd ‘𝑓)))⟩)
195	121, 122, 191, 193, 194	syl13anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st ‘𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd ‘𝑓)))⟩)
196	185, 189, 195	3eqtr4d 2788	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡) · 𝑓) = (𝑠 · ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩))
197	1, 2, 3, 4, 10, 19	dvhmulr 39100	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑠 × 𝑡) = (𝑠 ∘ 𝑡))
198	197	3adantr3 1170	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠 ∘ 𝑡))
199	198	oveq1d 7290	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠 ∘ 𝑡) · 𝑓))
200	138	oveq2d 7291	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 · 𝑓)) = (𝑠 · ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩))
201	196, 199, 200	3eqtr4d 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202		xp1st 7863	. . . . . . 7 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → (1st ‘𝑠) ∈ 𝑇)
203	202	adantl 482	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st ‘𝑠) ∈ 𝑇)
204		fvresi 7045	. . . . . 6 ⊢ ((1st ‘𝑠) ∈ 𝑇 → (( I ↾ 𝑇)‘(1st ‘𝑠)) = (1st ‘𝑠))
205	203, 204	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇)‘(1st ‘𝑠)) = (1st ‘𝑠))
206		xp2nd 7864	. . . . . . 7 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → (2nd ‘𝑠) ∈ 𝐸)
207	1, 2, 3	tendof 38777	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2nd ‘𝑠) ∈ 𝐸) → (2nd ‘𝑠):𝑇⟶𝑇)
208	206, 207	sylan2 593	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (2nd ‘𝑠):𝑇⟶𝑇)
209		fcoi2 6649	. . . . . 6 ⊢ ((2nd ‘𝑠):𝑇⟶𝑇 → (( I ↾ 𝑇) ∘ (2nd ‘𝑠)) = (2nd ‘𝑠))
210	208, 209	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∘ (2nd ‘𝑠)) = (2nd ‘𝑠))
211	205, 210	opeq12d 4812	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝑇)‘(1st ‘𝑠)), (( I ↾ 𝑇) ∘ (2nd ‘𝑠))⟩ = ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩)
212	1, 2, 3	tendoidcl 38783	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
213	212	anim1i 615	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ (𝑇 × 𝐸)))
214	1, 2, 3, 4, 12	dvhvsca 39115	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ (𝑇 × 𝐸))) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st ‘𝑠)), (( I ↾ 𝑇) ∘ (2nd ‘𝑠))⟩)
215	213, 214	syldan 591	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st ‘𝑠)), (( I ↾ 𝑇) ∘ (2nd ‘𝑠))⟩)
216		1st2nd2 7870	. . . . 5 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩)
217	216	adantl 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩)
218	211, 215, 217	3eqtr4d 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = 𝑠)
219	7, 9, 11, 13, 16, 18, 20, 26, 30, 34, 36, 120, 180, 201, 218	islmodd 20129	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod)
220	10	islvec 20366	. 2 ⊢ (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ 𝐷 ∈ DivRing))
221	219, 28, 220	sylanbrc 583	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   I cid 5488   × cxp 5587   ↾ cres 5591   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  inv_gcminusg 18578  1_rcur 19737  Ringcrg 19783  DivRingcdr 19991  LModclmod 20123  LVecclvec 20364  HLchlt 37364  LHypclh 37998  LTrncltrn 38115  TEndoctendo 38766  EDRingcedring 38767  DVecHcdvh 39092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-undef 8089  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-lmod 20125  df-lvec 20365  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-tendo 38769  df-edring 38771  df-dvech 39093

This theorem is referenced by:  dvhlvec  39123