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Theorem dvhlveclem 38745
Description: Lemma for dvhlvec 38746. 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 2738 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 38724 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2744 . . 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 2738 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
151, 3, 4, 10, 14dvhbase 38720 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
1615eqcomd 2744 . . 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 2738 . . . . . 6 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
221, 21, 4, 10dvhsca 38719 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
2322fveq2d 6678 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24 eqid 2738 . . . . 5 (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
251, 2, 21, 24erng1r 38632 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
2623, 25eqtr2d 2774 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) = (1r𝐷))
271, 21erngdv 38630 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2822, 27eqeltrd 2833 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
29 drngring 19628 . . . 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 38744 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 38740 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
36353impb 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡 ∈ (𝑇 × 𝐸)) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
37 simpl 486 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
38 simpr1 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
39 simpr2 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (𝑇 × 𝐸))
40 xp1st 7746 . . . . . . . 8 (𝑡 ∈ (𝑇 × 𝐸) → (1st𝑡) ∈ 𝑇)
4139, 40syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑡) ∈ 𝑇)
42 simpr3 1197 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
43 xp1st 7746 . . . . . . . 8 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
4442, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
451, 2, 3tendospdi1 38657 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (1st𝑡) ∈ 𝑇 ∧ (1st𝑓) ∈ 𝑇)) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
4637, 38, 41, 44, 45syl13anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
471, 2, 3, 4, 10, 8, 17dvhvadd 38729 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
48473adantr1 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
4948fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
50 fvex 6687 . . . . . . . . . 10 (1st𝑡) ∈ V
51 fvex 6687 . . . . . . . . . 10 (1st𝑓) ∈ V
5250, 51coex 7661 . . . . . . . . 9 ((1st𝑡) ∘ (1st𝑓)) ∈ V
53 ovex 7203 . . . . . . . . 9 ((2nd𝑡) (2nd𝑓)) ∈ V
5452, 53op1st 7722 . . . . . . . 8 (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((1st𝑡) ∘ (1st𝑓))
5549, 54eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st𝑡) ∘ (1st𝑓)))
5655fveq2d 6678 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = (𝑠‘((1st𝑡) ∘ (1st𝑓))))
571, 2, 3, 4, 12dvhvsca 38738 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
58573adantr3 1172 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
5958fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
60 fvex 6687 . . . . . . . . 9 (𝑠‘(1st𝑡)) ∈ V
61 vex 3402 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6687 . . . . . . . . . 10 (2nd𝑡) ∈ V
6361, 62coex 7661 . . . . . . . . 9 (𝑠 ∘ (2nd𝑡)) ∈ V
6460, 63op1st 7722 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠‘(1st𝑡))
6559, 64eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (𝑠‘(1st𝑡)))
661, 2, 3, 4, 12dvhvsca 38738 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
67663adantr2 1171 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
6867fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
69 fvex 6687 . . . . . . . . 9 (𝑠‘(1st𝑓)) ∈ V
70 fvex 6687 . . . . . . . . . 10 (2nd𝑓) ∈ V
7161, 70coex 7661 . . . . . . . . 9 (𝑠 ∘ (2nd𝑓)) ∈ V
7269, 71op1st 7722 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠‘(1st𝑓))
7368, 72eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
7465, 73coeq12d 5707 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
7546, 56, 743eqtr4d 2783 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))))
7630adantr 484 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
7716adantr 484 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
7838, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79 xp2nd 7747 . . . . . . . . . 10 (𝑡 ∈ (𝑇 × 𝐸) → (2nd𝑡) ∈ 𝐸)
8039, 79syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ 𝐸)
8180, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ (Base‘𝐷))
82 xp2nd 7747 . . . . . . . . . 10 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
8342, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
8483, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
8514, 17, 19ringdi 19438 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8676, 78, 81, 84, 85syl13anc 1373 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8714, 17ringacl 19450 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8876, 81, 84, 87syl3anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8988, 77eleqtrrd 2836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)
901, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
9137, 38, 89, 90syl12anc 836 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
921, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑡) ∈ 𝐸)) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
9337, 38, 80, 92syl12anc 836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
941, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9537, 38, 83, 94syl12anc 836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9693, 95oveq12d 7188 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9786, 91, 963eqtr3d 2781 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ ((2nd𝑡) (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9848fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
9952, 53op2nd 7723 . . . . . . . 8 (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((2nd𝑡) (2nd𝑓))
10098, 99eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd𝑡) (2nd𝑓)))
101100coeq2d 5705 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
10258fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
10360, 63op2nd 7723 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠 ∘ (2nd𝑡))
104102, 103eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd𝑡)))
10567fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
10669, 71op2nd 7723 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠 ∘ (2nd𝑓))
107105, 106eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
108104, 107oveq12d 7188 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
10997, 101, 1083eqtr4d 2783 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))))
11075, 109opeq12d 4769 . . . 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 38732 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1121113adantr1 1170 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1131, 2, 3, 4, 12dvhvsca 38738 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
11437, 38, 112, 113syl12anc 836 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
115353adantr3 1172 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
1161, 2, 3, 4, 12dvhvscacl 38740 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1171163adantr2 1171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1181, 2, 3, 4, 10, 8, 17dvhvadd 38729 . . . . 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 836 . . . 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 2783 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121 simpl 486 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
122 simpr1 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
123 simpr2 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡𝐸)
124 simpr3 1197 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
125124, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
126 eqid 2738 . . . . . . . 8 (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
1271, 2, 3, 21, 126erngplus2 38441 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇)) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
128121, 122, 123, 125, 127syl13anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
12922fveq2d 6678 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
13017, 129syl5eq 2785 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g‘((EDRing‘𝐾)‘𝑊)))
131130oveqd 7187 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑠 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132131fveq1d 6676 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝑠 𝑡)‘(1st𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)))
133132adantr 484 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡)‘(1st𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)))
134663adantr2 1171 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
135134fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
136135, 72eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
1371, 2, 3, 4, 12dvhvsca 38738 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
1381373adantr1 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
139138fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
140 fvex 6687 . . . . . . . . 9 (𝑡‘(1st𝑓)) ∈ V
141 vex 3402 . . . . . . . . . 10 𝑡 ∈ V
142141, 70coex 7661 . . . . . . . . 9 (𝑡 ∘ (2nd𝑓)) ∈ V
143140, 142op1st 7722 . . . . . . . 8 (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡‘(1st𝑓))
144139, 143eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st𝑓)))
145136, 144coeq12d 5707 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
146128, 133, 1453eqtr4d 2783 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡)‘(1st𝑓)) = ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))))
14730adantr 484 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
14816adantr 484 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
149122, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150123, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (Base‘𝐷))
151124, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
152151, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
15314, 17, 19ringdir 19439 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
154147, 149, 150, 152, 153syl13anc 1373 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
15514, 17ringacl 19450 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 𝑡) ∈ (Base‘𝐷))
156147, 149, 150, 155syl3anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ (Base‘𝐷))
157156, 148eleqtrrd 2836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
159121, 157, 151, 158syl12anc 836 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
160121, 122, 151, 94syl12anc 836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
1611, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
162121, 123, 151, 161syl12anc 836 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
163160, 162oveq12d 7188 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
164154, 159, 1633eqtr3d 2781 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
165134fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
166165, 106eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
167138fveq2d 6678 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
168140, 142op2nd 7723 . . . . . . . 8 (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡 ∘ (2nd𝑓))
169167, 168eqtrdi 2789 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd𝑓)))
170166, 169oveq12d 7188 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
171164, 170eqtr4d 2776 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))))
172146, 171opeq12d 4769 . . . 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 38738 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
174121, 157, 124, 173syl12anc 836 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
1751163adantr2 1171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1761, 2, 3, 4, 12dvhvscacl 38740 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1771763adantr1 1170 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1781, 2, 3, 4, 10, 8, 17dvhvadd 38729 . . . . 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 836 . . . 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 2783 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
1811, 2, 3tendocoval 38403 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸) ∧ (1st𝑓) ∈ 𝑇) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
182121, 122, 123, 125, 181syl121anc 1376 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
183 coass 6098 . . . . . . 7 ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))
184183a1i 11 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓))))
185182, 184opeq12d 4769 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
1861, 3tendococl 38409 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡𝐸) → (𝑠𝑡) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠𝑡) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 38738 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
189121, 187, 124, 188syl12anc 836 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
1901, 2, 3tendocl 38404 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇) → (𝑡‘(1st𝑓)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st𝑓)) ∈ 𝑇)
1921, 3tendococl 38409 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
1941, 2, 3, 4, 12dvhopvsca 38739 . . . . . 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 1373 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
196185, 189, 1953eqtr4d 2783 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
1971, 2, 3, 4, 10, 19dvhmulr 38723 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸)) → (𝑠 × 𝑡) = (𝑠𝑡))
1981973adantr3 1172 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠𝑡))
199198oveq1d 7185 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠𝑡) · 𝑓))
200138oveq2d 7186 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 · 𝑓)) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
201196, 199, 2003eqtr4d 2783 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202 xp1st 7746 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (1st𝑠) ∈ 𝑇)
203202adantl 485 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st𝑠) ∈ 𝑇)
204 fvresi 6945 . . . . . 6 ((1st𝑠) ∈ 𝑇 → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
205203, 204syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
206 xp2nd 7747 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (2nd𝑠) ∈ 𝐸)
2071, 2, 3tendof 38400 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (2nd𝑠) ∈ 𝐸) → (2nd𝑠):𝑇𝑇)
208206, 207sylan2 596 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (2nd𝑠):𝑇𝑇)
209 fcoi2 6553 . . . . . 6 ((2nd𝑠):𝑇𝑇 → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
210208, 209syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
211205, 210opeq12d 4769 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩ = ⟨(1st𝑠), (2nd𝑠)⟩)
2121, 2, 3tendoidcl 38406 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
213212anim1i 618 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸)))
2141, 2, 3, 4, 12dvhvsca 38738 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸))) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
215213, 214syldan 594 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
216 1st2nd2 7753 . . . . 5 (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
217216adantl 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
218211, 215, 2173eqtr4d 2783 . . 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 19759 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LMod)
22010islvec 19995 . 2 (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ 𝐷 ∈ DivRing))
221219, 28, 220sylanbrc 586 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  cop 4522   I cid 5428   × cxp 5523  cres 5527  ccom 5529  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  Basecbs 16586  +gcplusg 16668  .rcmulr 16669  Scalarcsca 16671   ·𝑠 cvsca 16672  0gc0g 16816  invgcminusg 18220  1rcur 19370  Ringcrg 19416  DivRingcdr 19621  LModclmod 19753  LVecclvec 19993  HLchlt 36987  LHypclh 37621  LTrncltrn 37738  TEndoctendo 38389  EDRingcedring 38390  DVecHcdvh 38715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-riotaBAD 36590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-tpos 7921  df-undef 7968  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-0g 16818  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-grp 18222  df-minusg 18223  df-mgp 19359  df-ur 19371  df-ring 19418  df-oppr 19495  df-dvdsr 19513  df-unit 19514  df-invr 19544  df-dvr 19555  df-drng 19623  df-lmod 19755  df-lvec 19994  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135  df-lplanes 37136  df-lvols 37137  df-lines 37138  df-psubsp 37140  df-pmap 37141  df-padd 37433  df-lhyp 37625  df-laut 37626  df-ldil 37741  df-ltrn 37742  df-trl 37796  df-tendo 38392  df-edring 38394  df-dvech 38716
This theorem is referenced by:  dvhlvec  38746
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