Theorem dvhlveclem 41127

Description: Lemma for dvhlvec 41128. 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 2735	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 41106	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2741	. . 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 2735	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	1, 3, 4, 10, 14	dvhbase 41102	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
16	15	eqcomd 2741	. . 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 2735	. . . . . 6 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
22	1, 21, 4, 10	dvhsca 41101	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
23	22	fveq2d 6880	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24		eqid 2735	. . . . 5 ⊢ (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
25	1, 2, 21, 24	erng1r 41014	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
26	23, 25	eqtr2d 2771	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) = (1r‘𝐷))
27	1, 21	erngdv 41012	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
28	22, 27	eqeltrd 2834	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
29		drngring 20696	. . . 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 41126	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)
35	1, 2, 3, 4, 12	dvhvscacl 41122	. . . 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 482	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38		simpr1 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ 𝐸)
39		simpr2 1196	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (𝑇 × 𝐸))
40		xp1st 8020	. . . . . . . 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 1197	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
43		xp1st 8020	. . . . . . . 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 41039	. . . . . . 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 1374	. . . . . 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 41111	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩)
48	47	3adantr1 1170	. . . . . . . . 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 6880	. . . . . . . 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 6889	. . . . . . . . . 10 ⊢ (1st ‘𝑡) ∈ V
51		fvex 6889	. . . . . . . . . 10 ⊢ (1st ‘𝑓) ∈ V
52	50, 51	coex 7926	. . . . . . . . 9 ⊢ ((1st ‘𝑡) ∘ (1st ‘𝑓)) ∈ V
53		ovex 7438	. . . . . . . . 9 ⊢ ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ V
54	52, 53	op1st 7996	. . . . . . . 8 ⊢ (1st ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩) = ((1st ‘𝑡) ∘ (1st ‘𝑓))
55	49, 54	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st ‘𝑡) ∘ (1st ‘𝑓)))
56	55	fveq2d 6880	. . . . . 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 41120	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩)
58	57	3adantr3 1172	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩)
59	58	fveq2d 6880	. . . . . . . 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 6889	. . . . . . . . 9 ⊢ (𝑠‘(1st ‘𝑡)) ∈ V
61		vex 3463	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
62		fvex 6889	. . . . . . . . . 10 ⊢ (2nd ‘𝑡) ∈ V
63	61, 62	coex 7926	. . . . . . . . 9 ⊢ (𝑠 ∘ (2nd ‘𝑡)) ∈ V
64	60, 63	op1st 7996	. . . . . . . 8 ⊢ (1st ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩) = (𝑠‘(1st ‘𝑡))
65	59, 64	eqtrdi 2786	. . . . . . 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 41120	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
67	66	3adantr2 1171	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
68	67	fveq2d 6880	. . . . . . . 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 6889	. . . . . . . . 9 ⊢ (𝑠‘(1st ‘𝑓)) ∈ V
70		fvex 6889	. . . . . . . . . 10 ⊢ (2nd ‘𝑓) ∈ V
71	61, 70	coex 7926	. . . . . . . . 9 ⊢ (𝑠 ∘ (2nd ‘𝑓)) ∈ V
72	69, 71	op1st 7996	. . . . . . . 8 ⊢ (1st ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩) = (𝑠‘(1st ‘𝑓))
73	68, 72	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st ‘𝑓)))
74	65, 73	coeq12d 5844	. . . . . 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 2780	. . . . 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 480	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
77	16	adantr 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
78	38, 77	eleqtrd 2836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79		xp2nd 8021	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑡) ∈ (Base‘𝐷))
82		xp2nd 8021	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ (Base‘𝐷))
85	14, 17, 19	ringdi 20221	. . . . . . . 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 1374	. . . . . . 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 20238	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Ring ∧ (2nd ‘𝑡) ∈ (Base‘𝐷) ∧ (2nd ‘𝑓) ∈ (Base‘𝐷)) → ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ (Base‘𝐷))
88	76, 81, 84, 87	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)) ∈ (Base‘𝐷))
89	88, 77	eleqtrrd 2837	. . . . . . . 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 41105	. . . . . . . 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 836	. . . . . . 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 41105	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd ‘𝑡) ∈ 𝐸)) → (𝑠 × (2nd ‘𝑡)) = (𝑠 ∘ (2nd ‘𝑡)))
93	37, 38, 80, 92	syl12anc 836	. . . . . . . 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 41105	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (𝑠 × (2nd ‘𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
95	37, 38, 83, 94	syl12anc 836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd ‘𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
96	93, 95	oveq12d 7423	. . . . . . 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 2778	. . . . . 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 6880	. . . . . . . 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 7997	. . . . . . . 8 ⊢ (2nd ‘⟨((1st ‘𝑡) ∘ (1st ‘𝑓)), ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))⟩) = ((2nd ‘𝑡) ⨣ (2nd ‘𝑓))
100	98, 99	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd ‘𝑡) ⨣ (2nd ‘𝑓)))
101	100	coeq2d 5842	. . . . . 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 6880	. . . . . . . 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 7997	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑠‘(1st ‘𝑡)), (𝑠 ∘ (2nd ‘𝑡))⟩) = (𝑠 ∘ (2nd ‘𝑡))
104	102, 103	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd ‘𝑡)))
105	67	fveq2d 6880	. . . . . . . 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 7997	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩) = (𝑠 ∘ (2nd ‘𝑓))
107	105, 106	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
108	104, 107	oveq12d 7423	. . . . . 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 2780	. . . . 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 4857	. . . 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 41114	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
112	111	3adantr1 1170	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
113	1, 2, 3, 4, 12	dvhvsca 41120	. . . . 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 836	. . . 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 1172	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
116	1, 2, 3, 4, 12	dvhvscacl 41122	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
117	116	3adantr2 1171	. . . . 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 41111	. . . . 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 836	. . . 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 2780	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121		simpl 482	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
122		simpr1 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ 𝐸)
123		simpr2 1196	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ 𝐸)
124		simpr3 1197	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
127	1, 2, 3, 21, 126	erngplus2 40823	. . . . . . 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 1374	. . . . . 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 6880	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
130	17, 129	eqtrid 2782	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (+g‘((EDRing‘𝐾)‘𝑊)))
131	130	oveqd 7422	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠 ⨣ 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132	131	fveq1d 6878	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑠 ⨣ 𝑡)‘(1st ‘𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st ‘𝑓)))
133	132	adantr 480	. . . . . 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 1171	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
135	134	fveq2d 6880	. . . . . . . 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 2786	. . . . . . 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 41120	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
138	137	3adantr1 1170	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
139	138	fveq2d 6880	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩))
140		fvex 6889	. . . . . . . . 9 ⊢ (𝑡‘(1st ‘𝑓)) ∈ V
141		vex 3463	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
142	141, 70	coex 7926	. . . . . . . . 9 ⊢ (𝑡 ∘ (2nd ‘𝑓)) ∈ V
143	140, 142	op1st 7996	. . . . . . . 8 ⊢ (1st ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = (𝑡‘(1st ‘𝑓))
144	139, 143	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st ‘𝑓)))
145	136, 144	coeq12d 5844	. . . . . 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 2780	. . . . 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 480	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
148	16	adantr 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
149	122, 148	eleqtrd 2836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150	123, 148	eleqtrd 2836	. . . . . . . 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 2836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘𝑓) ∈ (Base‘𝐷))
153	14, 17, 19	ringdir 20222	. . . . . . . 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 1374	. . . . . . 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 20238	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 ⨣ 𝑡) ∈ (Base‘𝐷))
156	147, 149, 150, 155	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ⨣ 𝑡) ∈ (Base‘𝐷))
157	156, 148	eleqtrrd 2837	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ⨣ 𝑡) ∈ 𝐸)
158	1, 2, 3, 4, 10, 19	dvhmulr 41105	. . . . . . . 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 836	. . . . . . 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 836	. . . . . . . 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 41105	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (𝑡 × (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
162	121, 123, 151, 161	syl12anc 836	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
163	160, 162	oveq12d 7423	. . . . . . 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 2778	. . . . . 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 6880	. . . . . . . 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 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd ‘𝑓)))
167	138	fveq2d 6880	. . . . . . . 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 7997	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩) = (𝑡 ∘ (2nd ‘𝑓))
169	167, 168	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
170	166, 169	oveq12d 7423	. . . . . 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 2773	. . . . 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 4857	. . . 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 41120	. . . . 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 836	. . . 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 1171	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
176	1, 2, 3, 4, 12	dvhvscacl 41122	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
177	176	3adantr1 1170	. . . . 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 41111	. . . . 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 836	. . . 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 2780	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ⨣ 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
181	1, 2, 3	tendocoval 40785	. . . . . . 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 1377	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 ∘ 𝑡)‘(1st ‘𝑓)) = (𝑠‘(𝑡‘(1st ‘𝑓))))
183		coass 6254	. . . . . . 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 4857	. . . . 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 40791	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠 ∘ 𝑡) ∈ 𝐸)
187	121, 122, 123, 186	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ 𝑡) ∈ 𝐸)
188	1, 2, 3, 4, 12	dvhvsca 41120	. . . . . 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 836	. . . . 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 40786	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ (1st ‘𝑓) ∈ 𝑇) → (𝑡‘(1st ‘𝑓)) ∈ 𝑇)
191	121, 123, 125, 190	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st ‘𝑓)) ∈ 𝑇)
192	1, 3	tendococl 40791	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ (2nd ‘𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd ‘𝑓)) ∈ 𝐸)
193	121, 123, 151, 192	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd ‘𝑓)) ∈ 𝐸)
194	1, 2, 3, 4, 12	dvhopvsca 41121	. . . . . 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 1374	. . . . 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 2780	. . . 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 41105	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑠 × 𝑡) = (𝑠 ∘ 𝑡))
198	197	3adantr3 1172	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠 ∘ 𝑡))
199	198	oveq1d 7420	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠 ∘ 𝑡) · 𝑓))
200	138	oveq2d 7421	. . . 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 2780	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202		xp1st 8020	. . . . . . 7 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → (1st ‘𝑠) ∈ 𝑇)
203	202	adantl 481	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st ‘𝑠) ∈ 𝑇)
204		fvresi 7165	. . . . . 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 8021	. . . . . . 7 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → (2nd ‘𝑠) ∈ 𝐸)
207	1, 2, 3	tendof 40782	. . . . . . 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 6753	. . . . . 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 4857	. . . 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 40788	. . . . . 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 41120	. . . . 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 8027	. . . . 5 ⊢ (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩)
217	216	adantl 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩)
218	211, 215, 217	3eqtr4d 2780	. . 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 20823	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LMod)
220	10	islvec 21062	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   I cid 5547   × cxp 5652   ↾ cres 5656   ∘ ccom 5658  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  +_gcplusg 17271  ._rcmulr 17272  Scalarcsca 17274   ·_𝑠 cvsca 17275  0_gc0g 17453  inv_gcminusg 18917  1_rcur 20141  Ringcrg 20193  DivRingcdr 20689  LModclmod 20817  LVecclvec 21060  HLchlt 39368  LHypclh 40003  LTrncltrn 40120  TEndoctendo 40771  EDRingcedring 40772  DVecHcdvh 41097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-riotaBAD 38971

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-undef 8272  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-0g 17455  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-p1 18436  df-lat 18442  df-clat 18509  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-dvr 20361  df-drng 20691  df-lmod 20819  df-lvec 21061  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39517  df-lplanes 39518  df-lvols 39519  df-lines 39520  df-psubsp 39522  df-pmap 39523  df-padd 39815  df-lhyp 40007  df-laut 40008  df-ldil 40123  df-ltrn 40124  df-trl 40178  df-tendo 40774  df-edring 40776  df-dvech 41098

This theorem is referenced by:  dvhlvec  41128