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Theorem dvhlveclem 35863
Description: Lemma for dvhlvec 35864. 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 2626 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 35842 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2632 . . 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 2626 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
151, 3, 4, 10, 14dvhbase 35838 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
1615eqcomd 2632 . . 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 2626 . . . . . 6 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
221, 21, 4, 10dvhsca 35837 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
2322fveq2d 6154 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r𝐷) = (1r‘((EDRing‘𝐾)‘𝑊)))
24 eqid 2626 . . . . 5 (1r‘((EDRing‘𝐾)‘𝑊)) = (1r‘((EDRing‘𝐾)‘𝑊))
251, 2, 21, 24erng1r 35749 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r‘((EDRing‘𝐾)‘𝑊)) = ( I ↾ 𝑇))
2623, 25eqtr2d 2661 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) = (1r𝐷))
271, 21erngdv 35747 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2822, 27eqeltrd 2704 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
29 drngring 18670 . . . 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 35862 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 35858 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
36353impb 1257 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡 ∈ (𝑇 × 𝐸)) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
37 simpl 473 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
38 simpr1 1065 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
39 simpr2 1066 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (𝑇 × 𝐸))
40 xp1st 7146 . . . . . . . 8 (𝑡 ∈ (𝑇 × 𝐸) → (1st𝑡) ∈ 𝑇)
4139, 40syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑡) ∈ 𝑇)
42 simpr3 1067 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
43 xp1st 7146 . . . . . . . 8 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
4442, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
451, 2, 3tendospdi1 35775 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (1st𝑡) ∈ 𝑇 ∧ (1st𝑓) ∈ 𝑇)) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
4637, 38, 41, 44, 45syl13anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘((1st𝑡) ∘ (1st𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
471, 2, 3, 4, 10, 8, 17dvhvadd 35847 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
48473adantr1 1218 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) = ⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩)
4948fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
50 fvex 6160 . . . . . . . . . 10 (1st𝑡) ∈ V
51 fvex 6160 . . . . . . . . . 10 (1st𝑓) ∈ V
5250, 51coex 7068 . . . . . . . . 9 ((1st𝑡) ∘ (1st𝑓)) ∈ V
53 ovex 6633 . . . . . . . . 9 ((2nd𝑡) (2nd𝑓)) ∈ V
5452, 53op1st 7124 . . . . . . . 8 (1st ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((1st𝑡) ∘ (1st𝑓))
5549, 54syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 + 𝑓)) = ((1st𝑡) ∘ (1st𝑓)))
5655fveq2d 6154 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠‘(1st ‘(𝑡 + 𝑓))) = (𝑠‘((1st𝑡) ∘ (1st𝑓))))
571, 2, 3, 4, 12dvhvsca 35856 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
58573adantr3 1220 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) = ⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩)
5958fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
60 fvex 6160 . . . . . . . . 9 (𝑠‘(1st𝑡)) ∈ V
61 vex 3194 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6160 . . . . . . . . . 10 (2nd𝑡) ∈ V
6361, 62coex 7068 . . . . . . . . 9 (𝑠 ∘ (2nd𝑡)) ∈ V
6460, 63op1st 7124 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠‘(1st𝑡))
6559, 64syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑡)) = (𝑠‘(1st𝑡)))
661, 2, 3, 4, 12dvhvsca 35856 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
67663adantr2 1219 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
6867fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
69 fvex 6160 . . . . . . . . 9 (𝑠‘(1st𝑓)) ∈ V
70 fvex 6160 . . . . . . . . . 10 (2nd𝑓) ∈ V
7161, 70coex 7068 . . . . . . . . 9 (𝑠 ∘ (2nd𝑓)) ∈ V
7269, 71op1st 7124 . . . . . . . 8 (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠‘(1st𝑓))
7368, 72syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
7465, 73coeq12d 5251 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑡)) ∘ (1st ‘(𝑠 · 𝑓))) = ((𝑠‘(1st𝑡)) ∘ (𝑠‘(1st𝑓))))
7546, 56, 743eqtr4d 2670 . . . . 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 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
79 xp2nd 7147 . . . . . . . . . 10 (𝑡 ∈ (𝑇 × 𝐸) → (2nd𝑡) ∈ 𝐸)
8039, 79syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ 𝐸)
8180, 77eleqtrd 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑡) ∈ (Base‘𝐷))
82 xp2nd 7147 . . . . . . . . . 10 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
8342, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
8483, 77eleqtrd 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
8514, 17, 19ringdi 18482 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8676, 78, 81, 84, 85syl13anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))))
8714, 17ringacl 18494 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ (2nd𝑡) ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8876, 81, 84, 87syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ (Base‘𝐷))
8988, 77eleqtrrd 2707 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)
901, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ ((2nd𝑡) (2nd𝑓)) ∈ 𝐸)) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
9137, 38, 89, 90syl12anc 1321 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × ((2nd𝑡) (2nd𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
921, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑡) ∈ 𝐸)) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
9337, 38, 80, 92syl12anc 1321 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑡)) = (𝑠 ∘ (2nd𝑡)))
941, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9537, 38, 83, 94syl12anc 1321 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
9693, 95oveq12d 6623 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑡)) (𝑠 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9786, 91, 963eqtr3d 2668 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ ((2nd𝑡) (2nd𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
9848fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩))
9952, 53op2nd 7125 . . . . . . . 8 (2nd ‘⟨((1st𝑡) ∘ (1st𝑓)), ((2nd𝑡) (2nd𝑓))⟩) = ((2nd𝑡) (2nd𝑓))
10098, 99syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 + 𝑓)) = ((2nd𝑡) (2nd𝑓)))
101100coeq2d 5249 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = (𝑠 ∘ ((2nd𝑡) (2nd𝑓))))
10258fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩))
10360, 63op2nd 7125 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑡)), (𝑠 ∘ (2nd𝑡))⟩) = (𝑠 ∘ (2nd𝑡))
104102, 103syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑡)) = (𝑠 ∘ (2nd𝑡)))
10567fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
10669, 71op2nd 7125 . . . . . . . 8 (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠 ∘ (2nd𝑓))
107105, 106syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
108104, 107oveq12d 6623 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))) = ((𝑠 ∘ (2nd𝑡)) (𝑠 ∘ (2nd𝑓))))
10997, 101, 1083eqtr4d 2670 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 ∘ (2nd ‘(𝑡 + 𝑓))) = ((2nd ‘(𝑠 · 𝑡)) (2nd ‘(𝑠 · 𝑓))))
11075, 109opeq12d 4383 . . . 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 35850 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1121113adantr1 1218 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))
1131, 2, 3, 4, 12dvhvsca 35856 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸 ∧ (𝑡 + 𝑓) ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
11437, 38, 112, 113syl12anc 1321 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ⟨(𝑠‘(1st ‘(𝑡 + 𝑓))), (𝑠 ∘ (2nd ‘(𝑡 + 𝑓)))⟩)
115353adantr3 1220 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑡) ∈ (𝑇 × 𝐸))
1161, 2, 3, 4, 12dvhvscacl 35858 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1171163adantr2 1219 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1181, 2, 3, 4, 10, 8, 17dvhvadd 35847 . . . . 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 1321 . . . 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 2670 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡 ∈ (𝑇 × 𝐸) ∧ 𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 + 𝑓)) = ((𝑠 · 𝑡) + (𝑠 · 𝑓)))
121 simpl 473 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
122 simpr1 1065 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠𝐸)
123 simpr2 1066 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡𝐸)
124 simpr3 1067 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑓 ∈ (𝑇 × 𝐸))
125124, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st𝑓) ∈ 𝑇)
126 eqid 2626 . . . . . . . 8 (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
1271, 2, 3, 21, 126erngplus2 35558 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇)) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
128121, 122, 123, 125, 127syl13anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡)‘(1st𝑓)) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
12922fveq2d 6154 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
13017, 129syl5eq 2672 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g‘((EDRing‘𝐾)‘𝑊)))
131130oveqd 6622 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑠 𝑡) = (𝑠(+g‘((EDRing‘𝐾)‘𝑊))𝑡))
132131fveq1d 6152 . . . . . . 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 1219 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
135134fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (1st ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
136135, 72syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑠 · 𝑓)) = (𝑠‘(1st𝑓)))
1371, 2, 3, 4, 12dvhvsca 35856 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
1381373adantr1 1218 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
139138fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
140 fvex 6160 . . . . . . . . 9 (𝑡‘(1st𝑓)) ∈ V
141 vex 3194 . . . . . . . . . 10 𝑡 ∈ V
142141, 70coex 7068 . . . . . . . . 9 (𝑡 ∘ (2nd𝑓)) ∈ V
143140, 142op1st 7124 . . . . . . . 8 (1st ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡‘(1st𝑓))
144139, 143syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (1st ‘(𝑡 · 𝑓)) = (𝑡‘(1st𝑓)))
145136, 144coeq12d 5251 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((1st ‘(𝑠 · 𝑓)) ∘ (1st ‘(𝑡 · 𝑓))) = ((𝑠‘(1st𝑓)) ∘ (𝑡‘(1st𝑓))))
146128, 133, 1453eqtr4d 2670 . . . . 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 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑠 ∈ (Base‘𝐷))
150123, 148eleqtrd 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → 𝑡 ∈ (Base‘𝐷))
151124, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ 𝐸)
152151, 148eleqtrd 2706 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd𝑓) ∈ (Base‘𝐷))
15314, 17, 19ringdir 18483 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷) ∧ (2nd𝑓) ∈ (Base‘𝐷))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
154147, 149, 150, 152, 153syl13anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))))
15514, 17ringacl 18494 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Base‘𝐷) ∧ 𝑡 ∈ (Base‘𝐷)) → (𝑠 𝑡) ∈ (Base‘𝐷))
156147, 149, 150, 155syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ (Base‘𝐷))
157156, 148eleqtrrd 2707 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 𝑡) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
159121, 157, 151, 158syl12anc 1321 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) × (2nd𝑓)) = ((𝑠 𝑡) ∘ (2nd𝑓)))
160121, 122, 151, 94syl12anc 1321 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × (2nd𝑓)) = (𝑠 ∘ (2nd𝑓)))
1611, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸)) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
162121, 123, 151, 161syl12anc 1321 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 × (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
163160, 162oveq12d 6623 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × (2nd𝑓)) (𝑡 × (2nd𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
164154, 159, 1633eqtr3d 2668 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
165134fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (2nd ‘⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
166165, 106syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑠 · 𝑓)) = (𝑠 ∘ (2nd𝑓)))
167138fveq2d 6154 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
168140, 142op2nd 7125 . . . . . . . 8 (2nd ‘⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = (𝑡 ∘ (2nd𝑓))
169167, 168syl6eq 2676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (2nd ‘(𝑡 · 𝑓)) = (𝑡 ∘ (2nd𝑓)))
170166, 169oveq12d 6623 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))) = ((𝑠 ∘ (2nd𝑓)) (𝑡 ∘ (2nd𝑓))))
171164, 170eqtr4d 2663 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) ∘ (2nd𝑓)) = ((2nd ‘(𝑠 · 𝑓)) (2nd ‘(𝑡 · 𝑓))))
172146, 171opeq12d 4383 . . . 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 35856 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠 𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
174121, 157, 124, 173syl12anc 1321 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ⟨((𝑠 𝑡)‘(1st𝑓)), ((𝑠 𝑡) ∘ (2nd𝑓))⟩)
1751163adantr2 1219 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · 𝑓) ∈ (𝑇 × 𝐸))
1761, 2, 3, 4, 12dvhvscacl 35858 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1771763adantr1 1218 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 · 𝑓) ∈ (𝑇 × 𝐸))
1781, 2, 3, 4, 10, 8, 17dvhvadd 35847 . . . . 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 1321 . . . 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 2670 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 𝑡) · 𝑓) = ((𝑠 · 𝑓) + (𝑡 · 𝑓)))
1811, 2, 3tendocoval 35520 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸) ∧ (1st𝑓) ∈ 𝑇) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
182121, 122, 123, 125, 181syl121anc 1328 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡)‘(1st𝑓)) = (𝑠‘(𝑡‘(1st𝑓))))
183 coass 5616 . . . . . . 7 ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))
184183a1i 11 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) ∘ (2nd𝑓)) = (𝑠 ∘ (𝑡 ∘ (2nd𝑓))))
185182, 184opeq12d 4383 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
1861, 3tendococl 35526 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑡𝐸) → (𝑠𝑡) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠𝑡) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 35856 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑠𝑡) ∈ 𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
189121, 187, 124, 188syl12anc 1321 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = ⟨((𝑠𝑡)‘(1st𝑓)), ((𝑠𝑡) ∘ (2nd𝑓))⟩)
1901, 2, 3tendocl 35521 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (1st𝑓) ∈ 𝑇) → (𝑡‘(1st𝑓)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡‘(1st𝑓)) ∈ 𝑇)
1921, 3tendococl 35526 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸 ∧ (2nd𝑓) ∈ 𝐸) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑡 ∘ (2nd𝑓)) ∈ 𝐸)
1941, 2, 3, 4, 12dvhopvsca 35857 . . . . . 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 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩) = ⟨(𝑠‘(𝑡‘(1st𝑓))), (𝑠 ∘ (𝑡 ∘ (2nd𝑓)))⟩)
196185, 189, 1953eqtr4d 2670 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠𝑡) · 𝑓) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
1971, 2, 3, 4, 10, 19dvhmulr 35841 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸)) → (𝑠 × 𝑡) = (𝑠𝑡))
1981973adantr3 1220 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 × 𝑡) = (𝑠𝑡))
199198oveq1d 6620 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = ((𝑠𝑡) · 𝑓))
200138oveq2d 6621 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → (𝑠 · (𝑡 · 𝑓)) = (𝑠 · ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩))
201196, 199, 2003eqtr4d 2670 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑡𝐸𝑓 ∈ (𝑇 × 𝐸))) → ((𝑠 × 𝑡) · 𝑓) = (𝑠 · (𝑡 · 𝑓)))
202 xp1st 7146 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (1st𝑠) ∈ 𝑇)
203202adantl 482 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (1st𝑠) ∈ 𝑇)
204 tendospid 35772 . . . . . 6 ((1st𝑠) ∈ 𝑇 → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
205203, 204syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇)‘(1st𝑠)) = (1st𝑠))
206 xp2nd 7147 . . . . . . 7 (𝑠 ∈ (𝑇 × 𝐸) → (2nd𝑠) ∈ 𝐸)
2071, 2, 3tendof 35517 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (2nd𝑠) ∈ 𝐸) → (2nd𝑠):𝑇𝑇)
208206, 207sylan2 491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (2nd𝑠):𝑇𝑇)
209 fcoi2 6038 . . . . . 6 ((2nd𝑠):𝑇𝑇 → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
210208, 209syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∘ (2nd𝑠)) = (2nd𝑠))
211205, 210opeq12d 4383 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩ = ⟨(1st𝑠), (2nd𝑠)⟩)
2121, 2, 3tendoidcl 35523 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
213212anim1i 591 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸)))
2141, 2, 3, 4, 12dvhvsca 35856 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸𝑠 ∈ (𝑇 × 𝐸))) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
215213, 214syldan 487 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝑇) · 𝑠) = ⟨(( I ↾ 𝑇)‘(1st𝑠)), (( I ↾ 𝑇) ∘ (2nd𝑠))⟩)
216 1st2nd2 7153 . . . . 5 (𝑠 ∈ (𝑇 × 𝐸) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
217216adantl 482 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ (𝑇 × 𝐸)) → 𝑠 = ⟨(1st𝑠), (2nd𝑠)⟩)
218211, 215, 2173eqtr4d 2670 . . 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 18785 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LMod)
22010islvec 19018 . 2 (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ 𝐷 ∈ DivRing))
221219, 28, 220sylanbrc 697 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  cop 4159   I cid 4989   × cxp 5077  cres 5081  ccom 5083  wf 5846  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  Basecbs 15776  +gcplusg 15857  .rcmulr 15858  Scalarcsca 15860   ·𝑠 cvsca 15861  0gc0g 16016  invgcminusg 17339  1rcur 18417  Ringcrg 18463  DivRingcdr 18663  LModclmod 18779  LVecclvec 19016  HLchlt 34103  LHypclh 34736  LTrncltrn 34853  TEndoctendo 35506  EDRingcedring 35507  DVecHcdvh 35833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-riotaBAD 33705
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-tpos 7298  df-undef 7345  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-0g 16018  df-preset 16844  df-poset 16862  df-plt 16874  df-lub 16890  df-glb 16891  df-join 16892  df-meet 16893  df-p0 16955  df-p1 16956  df-lat 16962  df-clat 17024  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-mgp 18406  df-ur 18418  df-ring 18465  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-dvr 18599  df-drng 18665  df-lmod 18781  df-lvec 19017  df-oposet 33929  df-ol 33931  df-oml 33932  df-covers 34019  df-ats 34020  df-atl 34051  df-cvlat 34075  df-hlat 34104  df-llines 34250  df-lplanes 34251  df-lvols 34252  df-lines 34253  df-psubsp 34255  df-pmap 34256  df-padd 34548  df-lhyp 34740  df-laut 34741  df-ldil 34856  df-ltrn 34857  df-trl 34912  df-tendo 35509  df-edring 35511  df-dvech 35834
This theorem is referenced by:  dvhlvec  35864
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