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Theorem dvhlveclem 41110
Description: Lemma for dvhlvec 41111. 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 2737 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 41089 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2743 . . 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 2737 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
151, 3, 4, 10, 14dvhbase 41085 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
1615eqcomd 2743 . . 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 2737 . . . . . 6 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
221, 21, 4, 10dvhsca 41084 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
2322fveq2d 6910 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24 eqid 2737 . . . . 5 (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
251, 2, 21, 24erng1r 40997 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
2623, 25eqtr2d 2778 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) = (1r𝐷))
271, 21erngdv 40995 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2822, 27eqeltrd 2841 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
29 drngring 20736 . . . 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 41109 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 41105 . . . 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 482 . . . . . . 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 8046 . . . . . . . 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 8046 . . . . . . . 8 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
4442, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
451, 2, 3tendospdi1 41022 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (1st𝑡) ∈ 𝑇 ∧ (1st𝑓) ∈ 𝑇)) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
4637, 38, 41, 44, 45syl13anc 1374 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
471, 2, 3, 4, 10, 8, 17dvhvadd 41094 . . . . . . . . . 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 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
50 fvex 6919 . . . . . . . . . 10 (1st𝑡) ∈ V
51 fvex 6919 . . . . . . . . . 10 (1st𝑓) ∈ V
5250, 51coex 7952 . . . . . . . . 9 ((1st𝑡) ∘ (1st𝑓)) ∈ V
53 ovex 7464 . . . . . . . . 9 ((2nd𝑡) (2nd𝑓)) ∈ V
5452, 53op1st 8022 . . . . . . . 8 (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((1st𝑡) ∘ (1st𝑓))
5549, 54eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st𝑡) ∘ (1st𝑓)))
5655fveq2d 6910 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = (𝑠‘((1st𝑡) ∘ (1st𝑓))))
571, 2, 3, 4, 12dvhvsca 41103 . . . . . . . . . 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 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
60 fvex 6919 . . . . . . . . 9 (𝑠‘(1st𝑡)) ∈ V
61 vex 3484 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6919 . . . . . . . . . 10 (2nd𝑡) ∈ V
6361, 62coex 7952 . . . . . . . . 9 (𝑠 ∘ (2nd𝑡)) ∈ V
6460, 63op1st 8022 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠‘(1st𝑡))
6559, 64eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (𝑠‘(1st𝑡)))
661, 2, 3, 4, 12dvhvsca 41103 . . . . . . . . . 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 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
69 fvex 6919 . . . . . . . . 9 (𝑠‘(1st𝑓)) ∈ V
70 fvex 6919 . . . . . . . . . 10 (2nd𝑓) ∈ V
7161, 70coex 7952 . . . . . . . . 9 (𝑠 ∘ (2nd𝑓)) ∈ V
7269, 71op1st 8022 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠‘(1st𝑓))
7368, 72eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
7465, 73coeq12d 5875 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
7546, 56, 743eqtr4d 2787 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))))
7630adantr 480 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
7716adantr 480 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
7838, 77eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79 xp2nd 8047 . . . . . . . . . 10 (𝑡 ∈ (𝑇 × 𝐸) → (2nd𝑡) ∈ 𝐸)
8039, 79syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ 𝐸)
8180, 77eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ (Base‘𝐷))
82 xp2nd 8047 . . . . . . . . . 10 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
8342, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
8483, 77eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
8514, 17, 19ringdi 20258 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8676, 78, 81, 84, 85syl13anc 1374 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8714, 17ringacl 20275 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8876, 81, 84, 87syl3anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8988, 77eleqtrrd 2844 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)
901, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
9137, 38, 89, 90syl12anc 837 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
921, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑡) ∈ 𝐸)) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
9337, 38, 80, 92syl12anc 837 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
941, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9537, 38, 83, 94syl12anc 837 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9693, 95oveq12d 7449 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9786, 91, 963eqtr3d 2785 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ ((2nd𝑡) (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9848fveq2d 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
9952, 53op2nd 8023 . . . . . . . 8 (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((2nd𝑡) (2nd𝑓))
10098, 99eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd𝑡) (2nd𝑓)))
101100coeq2d 5873 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
10258fveq2d 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
10360, 63op2nd 8023 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠 ∘ (2nd𝑡))
104102, 103eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd𝑡)))
10567fveq2d 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
10669, 71op2nd 8023 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠 ∘ (2nd𝑓))
107105, 106eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
108104, 107oveq12d 7449 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
10997, 101, 1083eqtr4d 2787 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))))
11075, 109opeq12d 4881 . . . 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 41097 . . . . . 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 41103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
11437, 38, 112, 113syl12anc 837 . . . 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 41105 . . . . . 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 41094 . . . . 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 837 . . . 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 2787 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121 simpl 482 . . . . . . 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 2737 . . . . . . . 8 (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
1271, 2, 3, 21, 126erngplus2 40806 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇)) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
128121, 122, 123, 125, 127syl13anc 1374 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
12922fveq2d 6910 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
13017, 129eqtrid 2789 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g‘((EDRing‘𝐾)‘𝑊)))
131130oveqd 7448 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑠 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132131fveq1d 6908 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝑠 𝑡)‘(1st𝑓)) = ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)))
133132adantr 480 . . . . . 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 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
136135, 72eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
1371, 2, 3, 4, 12dvhvsca 41103 . . . . . . . . . 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 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
140 fvex 6919 . . . . . . . . 9 (𝑡‘(1st𝑓)) ∈ V
141 vex 3484 . . . . . . . . . 10 𝑡 ∈ V
142141, 70coex 7952 . . . . . . . . 9 (𝑡 ∘ (2nd𝑓)) ∈ V
143140, 142op1st 8022 . . . . . . . 8 (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡‘(1st𝑓))
144139, 143eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st𝑓)))
145136, 144coeq12d 5875 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
146128, 133, 1453eqtr4d 2787 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡)‘(1st𝑓)) = ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))))
14730adantr 480 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐷 ∈ Ring)
14816adantr 480 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝐸 = (Base‘𝐷))
149122, 148eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150123, 148eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (Base‘𝐷))
151124, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
152151, 148eleqtrd 2843 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
15314, 17, 19ringdir 20259 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
154147, 149, 150, 152, 153syl13anc 1374 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
15514, 17ringacl 20275 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 𝑡) ∈ (Base‘𝐷))
156147, 149, 150, 155syl3anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ (Base‘𝐷))
157156, 148eleqtrrd 2844 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
159121, 157, 151, 158syl12anc 837 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
160121, 122, 151, 94syl12anc 837 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
1611, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
162121, 123, 151, 161syl12anc 837 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
163160, 162oveq12d 7449 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
164154, 159, 1633eqtr3d 2785 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
165134fveq2d 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
166165, 106eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
167138fveq2d 6910 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
168140, 142op2nd 8023 . . . . . . . 8 (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡 ∘ (2nd𝑓))
169167, 168eqtrdi 2793 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd𝑓)))
170166, 169oveq12d 7449 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
171164, 170eqtr4d 2780 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))))
172146, 171opeq12d 4881 . . . 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 41103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
174121, 157, 124, 173syl12anc 837 . . . 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 41105 . . . . . 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 41094 . . . . 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 837 . . . 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 2787 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
1811, 2, 3tendocoval 40768 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸) ∧ (1st𝑓) ∈ 𝑇) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
182121, 122, 123, 125, 181syl121anc 1377 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
183 coass 6285 . . . . . . 7 ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))
184183a1i 11 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓))))
185182, 184opeq12d 4881 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
1861, 3tendococl 40774 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡𝐸) → (𝑠𝑡) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠𝑡) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 41103 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
189121, 187, 124, 188syl12anc 837 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
1901, 2, 3tendocl 40769 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇) → (𝑡‘(1st𝑓)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st𝑓)) ∈ 𝑇)
1921, 3tendococl 40774 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
1941, 2, 3, 4, 12dvhopvsca 41104 . . . . . 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 1374 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
196185, 189, 1953eqtr4d 2787 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
1971, 2, 3, 4, 10, 19dvhmulr 41088 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸)) → (𝑠 × 𝑡) = (𝑠𝑡))
1981973adantr3 1172 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠𝑡))
199198oveq1d 7446 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠𝑡) · 𝑓))
200138oveq2d 7447 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 · 𝑓)) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
201196, 199, 2003eqtr4d 2787 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202 xp1st 8046 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (1st𝑠) ∈ 𝑇)
203202adantl 481 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st𝑠) ∈ 𝑇)
204 fvresi 7193 . . . . . 6 ((1st𝑠) ∈ 𝑇 → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
205203, 204syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
206 xp2nd 8047 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (2nd𝑠) ∈ 𝐸)
2071, 2, 3tendof 40765 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (2nd𝑠) ∈ 𝐸) → (2nd𝑠):𝑇𝑇)
208206, 207sylan2 593 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (2nd𝑠):𝑇𝑇)
209 fcoi2 6783 . . . . . 6 ((2nd𝑠):𝑇𝑇 → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
210208, 209syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
211205, 210opeq12d 4881 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩ = ⟨(1st𝑠), (2nd𝑠)⟩)
2121, 2, 3tendoidcl 40771 . . . . . 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 41103 . . . . 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 8053 . . . . 5 (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
217216adantl 481 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
218211, 215, 2173eqtr4d 2787 . . 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 20864 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LMod)
22010islvec 21103 . 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 395  w3a 1087   = wceq 1540  wcel 2108  cop 4632   I cid 5577   × cxp 5683  cres 5687  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  invgcminusg 18952  1rcur 20178  Ringcrg 20230  DivRingcdr 20729  LModclmod 20858  LVecclvec 21101  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  TEndoctendo 40754  EDRingcedring 40755  DVecHcdvh 41080
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-undef 8298  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-drng 20731  df-lmod 20860  df-lvec 21102  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tendo 40757  df-edring 40759  df-dvech 41081
This theorem is referenced by:  dvhlvec  41111
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