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Theorem dvhlveclem 39571
Description: Lemma for dvhlvec 39572. 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 = (+g𝐷)
dvhgrp.a + = (+g𝑈)
dvhgrp.o 0 = (0g𝐷)
dvhgrp.i 𝐼 = (invg𝐷)
dvhlvec.m × = (.r𝐷)
dvhlvec.s · = ( ·𝑠𝑈)
Assertion
Ref Expression
dvhlveclem ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)

Proof of Theorem dvhlveclem
Dummy variables 𝑡 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef 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 2736 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 39550 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2742 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑇 × 𝐸) = (Base‘𝑈))
8 dvhgrp.a . . . 4 + = (+g𝑈)
98a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → + = (+g𝑈))
10 dvhgrp.d . . . 4 𝐷 = (Scalar‘𝑈)
1110a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = (Scalar‘𝑈))
12 dvhlvec.s . . . 4 · = ( ·𝑠𝑈)
1312a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → · = ( ·𝑠𝑈))
14 eqid 2736 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
151, 3, 4, 10, 14dvhbase 39546 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
1615eqcomd 2742 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐸 = (Base‘𝐷))
17 dvhgrp.p . . . 4 = (+g𝐷)
1817a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g𝐷))
19 dvhlvec.m . . . 4 × = (.r𝐷)
2019a1i 11 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → × = (.r𝐷))
21 eqid 2736 . . . . . 6 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
221, 21, 4, 10dvhsca 39545 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
2322fveq2d 6846 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24 eqid 2736 . . . . 5 (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
251, 2, 21, 24erng1r 39458 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
2623, 25eqtr2d 2777 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) = (1r𝐷))
271, 21erngdv 39456 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2822, 27eqeltrd 2838 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
29 drngring 20192 . . . 4 (𝐷 ∈ DivRing → 𝐷 ∈ Ring)
3028, 29syl 17 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)
31 dvhgrp.b . . . 4 𝐵 = (Base‘𝐾)
32 dvhgrp.o . . . 4 0 = (0g𝐷)
33 dvhgrp.i . . . 4 𝐼 = (invg𝐷)
3431, 1, 2, 3, 4, 10, 17, 8, 32, 33dvhgrp 39570 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 39566 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
36353impb 1115 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡 ∈ (𝑇 × 𝐸)) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
37 simpl 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
38 simpr1 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
39 simpr2 1195 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (𝑇 × 𝐸))
40 xp1st 7953 . . . . . . . 8 (𝑡 ∈ (𝑇 × 𝐸) → (1st𝑡) ∈ 𝑇)
4139, 40syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑡) ∈ 𝑇)
42 simpr3 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
43 xp1st 7953 . . . . . . . 8 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
4442, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
451, 2, 3tendospdi1 39483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (1st𝑡) ∈ 𝑇 ∧ (1st𝑓) ∈ 𝑇)) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
4637, 38, 41, 44, 45syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
471, 2, 3, 4, 10, 8, 17dvhvadd 39555 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
48473adantr1 1169 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
4948fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
50 fvex 6855 . . . . . . . . . 10 (1st𝑡) ∈ V
51 fvex 6855 . . . . . . . . . 10 (1st𝑓) ∈ V
5250, 51coex 7867 . . . . . . . . 9 ((1st𝑡) ∘ (1st𝑓)) ∈ V
53 ovex 7390 . . . . . . . . 9 ((2nd𝑡) (2nd𝑓)) ∈ V
5452, 53op1st 7929 . . . . . . . 8 (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((1st𝑡) ∘ (1st𝑓))
5549, 54eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st𝑡) ∘ (1st𝑓)))
5655fveq2d 6846 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = (𝑠‘((1st𝑡) ∘ (1st𝑓))))
571, 2, 3, 4, 12dvhvsca 39564 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
58573adantr3 1171 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
5958fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
60 fvex 6855 . . . . . . . . 9 (𝑠‘(1st𝑡)) ∈ V
61 vex 3449 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6855 . . . . . . . . . 10 (2nd𝑡) ∈ V
6361, 62coex 7867 . . . . . . . . 9 (𝑠 ∘ (2nd𝑡)) ∈ V
6460, 63op1st 7929 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠‘(1st𝑡))
6559, 64eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (𝑠‘(1st𝑡)))
661, 2, 3, 4, 12dvhvsca 39564 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
67663adantr2 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
6867fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
69 fvex 6855 . . . . . . . . 9 (𝑠‘(1st𝑓)) ∈ V
70 fvex 6855 . . . . . . . . . 10 (2nd𝑓) ∈ V
7161, 70coex 7867 . . . . . . . . 9 (𝑠 ∘ (2nd𝑓)) ∈ V
7269, 71op1st 7929 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠‘(1st𝑓))
7368, 72eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
7465, 73coeq12d 5820 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
7546, 56, 743eqtr4d 2786 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))))
7630adantr 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
7716adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
7838, 77eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79 xp2nd 7954 . . . . . . . . . 10 (𝑡 ∈ (𝑇 × 𝐸) → (2nd𝑡) ∈ 𝐸)
8039, 79syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ 𝐸)
8180, 77eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ (Base‘𝐷))
82 xp2nd 7954 . . . . . . . . . 10 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
8342, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
8483, 77eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
8514, 17, 19ringdi 19987 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8676, 78, 81, 84, 85syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8714, 17ringacl 19999 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8876, 81, 84, 87syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8988, 77eleqtrrd 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)
901, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
9137, 38, 89, 90syl12anc 835 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
921, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑡) ∈ 𝐸)) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
9337, 38, 80, 92syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
941, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9537, 38, 83, 94syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9693, 95oveq12d 7375 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9786, 91, 963eqtr3d 2784 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ ((2nd𝑡) (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9848fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
9952, 53op2nd 7930 . . . . . . . 8 (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((2nd𝑡) (2nd𝑓))
10098, 99eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd𝑡) (2nd𝑓)))
101100coeq2d 5818 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
10258fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
10360, 63op2nd 7930 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠 ∘ (2nd𝑡))
104102, 103eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd𝑡)))
10567fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
10669, 71op2nd 7930 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠 ∘ (2nd𝑓))
107105, 106eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
108104, 107oveq12d 7375 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
10997, 101, 1083eqtr4d 2786 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))))
11075, 109opeq12d 4838 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (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)))⟩)
1111, 2, 3, 4, 10, 17, 8dvhvaddcl 39558 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1121113adantr1 1169 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1131, 2, 3, 4, 12dvhvsca 39564 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
11437, 38, 112, 113syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
115353adantr3 1171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
1161, 2, 3, 4, 12dvhvscacl 39566 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1171163adantr2 1170 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1181, 2, 3, 4, 10, 8, 17dvhvadd 39555 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 · 𝑡) ∈ (𝑇 × 𝐸) ∧ (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑡) + (𝑠 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))), ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓)))⟩)
11937, 115, 117, 118syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑡) + (𝑠 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))), ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓)))⟩)
120110, 114, 1193eqtr4d 2786 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121 simpl 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
122 simpr1 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
123 simpr2 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡𝐸)
124 simpr3 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
125124, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
126 eqid 2736 . . . . . . . 8 (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
1271, 2, 3, 21, 126erngplus2 39267 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇)) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
128121, 122, 123, 125, 127syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
12922fveq2d 6846 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
13017, 129eqtrid 2788 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g‘((EDRing‘𝐾)‘𝑊)))
131130oveqd 7374 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑠 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132131fveq1d 6844 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝑠 𝑡)‘(1st𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)))
133132adantr 481 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡)‘(1st𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)))
134663adantr2 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
135134fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
136135, 72eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
1371, 2, 3, 4, 12dvhvsca 39564 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
1381373adantr1 1169 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
139138fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
140 fvex 6855 . . . . . . . . 9 (𝑡‘(1st𝑓)) ∈ V
141 vex 3449 . . . . . . . . . 10 𝑡 ∈ V
142141, 70coex 7867 . . . . . . . . 9 (𝑡 ∘ (2nd𝑓)) ∈ V
143140, 142op1st 7929 . . . . . . . 8 (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡‘(1st𝑓))
144139, 143eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st𝑓)))
145136, 144coeq12d 5820 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
146128, 133, 1453eqtr4d 2786 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡)‘(1st𝑓)) = ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))))
14730adantr 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
14816adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
149122, 148eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150123, 148eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (Base‘𝐷))
151124, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
152151, 148eleqtrd 2840 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
15314, 17, 19ringdir 19988 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
154147, 149, 150, 152, 153syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
15514, 17ringacl 19999 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 𝑡) ∈ (Base‘𝐷))
156147, 149, 150, 155syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ (Base‘𝐷))
157156, 148eleqtrrd 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
159121, 157, 151, 158syl12anc 835 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
160121, 122, 151, 94syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
1611, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
162121, 123, 151, 161syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
163160, 162oveq12d 7375 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
164154, 159, 1633eqtr3d 2784 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
165134fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
166165, 106eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
167138fveq2d 6846 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
168140, 142op2nd 7930 . . . . . . . 8 (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡 ∘ (2nd𝑓))
169167, 168eqtrdi 2792 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd𝑓)))
170166, 169oveq12d 7375 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
171164, 170eqtr4d 2779 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))))
172146, 171opeq12d 4838 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩ = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓)))⟩)
1731, 2, 3, 4, 12dvhvsca 39564 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
174121, 157, 124, 173syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
1751163adantr2 1170 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1761, 2, 3, 4, 12dvhvscacl 39566 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1771763adantr1 1169 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1781, 2, 3, 4, 10, 8, 17dvhvadd 39555 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 · 𝑓) ∈ (𝑇 × 𝐸) ∧ (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑓) + (𝑡 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓)))⟩)
179121, 175, 177, 178syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 · 𝑓) + (𝑡 · 𝑓)) = ⟨((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))), ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓)))⟩)
180172, 174, 1793eqtr4d 2786 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
1811, 2, 3tendocoval 39229 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸) ∧ (1st𝑓) ∈ 𝑇) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
182121, 122, 123, 125, 181syl121anc 1375 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
183 coass 6217 . . . . . . 7 ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))
184183a1i 11 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓))))
185182, 184opeq12d 4838 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
1861, 3tendococl 39235 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡𝐸) → (𝑠𝑡) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠𝑡) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 39564 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
189121, 187, 124, 188syl12anc 835 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
1901, 2, 3tendocl 39230 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇) → (𝑡‘(1st𝑓)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st𝑓)) ∈ 𝑇)
1921, 3tendococl 39235 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
1941, 2, 3, 4, 12dvhopvsca 39565 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (𝑡‘(1st𝑓)) ∈ 𝑇 ∧ (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)) → (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
195121, 122, 191, 193, 194syl13anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
196185, 189, 1953eqtr4d 2786 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
1971, 2, 3, 4, 10, 19dvhmulr 39549 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸)) → (𝑠 × 𝑡) = (𝑠𝑡))
1981973adantr3 1171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠𝑡))
199198oveq1d 7372 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠𝑡) · 𝑓))
200138oveq2d 7373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 · 𝑓)) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
201196, 199, 2003eqtr4d 2786 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202 xp1st 7953 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (1st𝑠) ∈ 𝑇)
203202adantl 482 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st𝑠) ∈ 𝑇)
204 fvresi 7119 . . . . . 6 ((1st𝑠) ∈ 𝑇 → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
205203, 204syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
206 xp2nd 7954 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (2nd𝑠) ∈ 𝐸)
2071, 2, 3tendof 39226 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (2nd𝑠) ∈ 𝐸) → (2nd𝑠):𝑇𝑇)
208206, 207sylan2 593 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (2nd𝑠):𝑇𝑇)
209 fcoi2 6717 . . . . . 6 ((2nd𝑠):𝑇𝑇 → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
210208, 209syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
211205, 210opeq12d 4838 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩ = ⟨(1st𝑠), (2nd𝑠)⟩)
2121, 2, 3tendoidcl 39232 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
213212anim1i 615 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸)))
2141, 2, 3, 4, 12dvhvsca 39564 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸))) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
215213, 214syldan 591 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
216 1st2nd2 7960 . . . . 5 (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
217216adantl 482 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
218211, 215, 2173eqtr4d 2786 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = 𝑠)
2197, 9, 11, 13, 16, 18, 20, 26, 30, 34, 36, 120, 180, 201, 218islmodd 20328 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LMod)
22010islvec 20565 . 2 (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ 𝐷 ∈ DivRing))
221219, 28, 220sylanbrc 583 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cop 4592   I cid 5530   × cxp 5631  cres 5635  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321  invgcminusg 18749  1rcur 19913  Ringcrg 19964  DivRingcdr 20185  LModclmod 20322  LVecclvec 20563  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  TEndoctendo 39215  EDRingcedring 39216  DVecHcdvh 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lvec 20564  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tendo 39218  df-edring 39220  df-dvech 39542
This theorem is referenced by:  dvhlvec  39572
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