Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diblss Structured version   Visualization version   GIF version

Theorem diblss 41152
Description: The value of partial isomorphism B is a subspace of partial vector space H. TODO: use dib* specific theorems instead of dia* ones to shorten proof? (Contributed by NM, 11-Feb-2014.)
Hypotheses
Ref Expression
diblss.b 𝐵 = (Base‘𝐾)
diblss.l = (le‘𝐾)
diblss.h 𝐻 = (LHyp‘𝐾)
diblss.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
diblss.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
diblss.s 𝑆 = (LSubSp‘𝑈)
Assertion
Ref Expression
diblss (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)

Proof of Theorem diblss
Dummy variables 𝑎 𝑏 𝑥 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2735 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2 diblss.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 eqid 2734 . . . . 5 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4 diblss.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 eqid 2734 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2734 . . . . 5 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
72, 3, 4, 5, 6dvhbase 41065 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
87eqcomd 2740 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
98adantr 480 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
10 eqid 2734 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11 eqid 2734 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
122, 10, 3, 4, 11dvhvbase 41069 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1312eqcomd 2740 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
1413adantr 480 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
15 eqidd 2735 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (+g𝑈) = (+g𝑈))
16 eqidd 2735 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ( ·𝑠𝑈) = ( ·𝑠𝑈))
17 diblss.s . . 3 𝑆 = (LSubSp‘𝑈)
1817a1i 11 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑆 = (LSubSp‘𝑈))
19 diblss.b . . . 4 𝐵 = (Base‘𝐾)
20 diblss.l . . . 4 = (le‘𝐾)
21 diblss.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
2219, 20, 2, 21, 4, 11dibss 41151 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (Base‘𝑈))
2322, 14sseqtrrd 4036 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
2419, 20, 2, 21dibn0 41135 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
25 fvex 6919 . . . . . . 7 (𝑥‘(1st𝑎)) ∈ V
26 vex 3481 . . . . . . . 8 𝑥 ∈ V
27 fvex 6919 . . . . . . . 8 (2nd𝑎) ∈ V
2826, 27coex 7952 . . . . . . 7 (𝑥 ∘ (2nd𝑎)) ∈ V
2925, 28op1st 8020 . . . . . 6 (1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥‘(1st𝑎))
3029coeq1i 5872 . . . . 5 ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) = ((𝑥‘(1st𝑎)) ∘ (1st𝑏))
31 simpll 767 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
32 simpr1 1193 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
33 simplr 769 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑋𝐵𝑋 𝑊))
34 simpr2 1194 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (𝐼𝑋))
3519, 20, 2, 10, 21dibelval1st1 41132 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
3631, 33, 34, 35syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
372, 10, 3tendocl 40749 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
3831, 32, 36, 37syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
39 simpr3 1195 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (𝐼𝑋))
4019, 20, 2, 10, 21dibelval1st1 41132 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
4131, 33, 39, 40syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
422, 10ltrnco 40701 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
4331, 38, 41, 42syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
44 simplll 775 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ HL)
4544hllatd 39345 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ Lat)
46 eqid 2734 . . . . . . . . 9 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
4719, 2, 10, 46trlcl 40146 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ∈ 𝐵)
4831, 43, 47syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ∈ 𝐵)
4919, 2, 10, 46trlcl 40146 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5031, 38, 49syl2anc 584 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5119, 2, 10, 46trlcl 40146 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
5231, 41, 51syl2anc 584 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
53 eqid 2734 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
5419, 53latjcl 18496 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵 ∧ (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))) ∈ 𝐵)
5545, 50, 52, 54syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))) ∈ 𝐵)
56 simplrl 777 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑋𝐵)
5720, 53, 2, 10, 46trlco 40709 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))))
5831, 38, 41, 57syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))))
5919, 2, 10, 46trlcl 40146 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6031, 36, 59syl2anc 584 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6120, 2, 10, 46, 3tendotp 40743 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) (((trL‘𝐾)‘𝑊)‘(1st𝑎)))
6231, 32, 36, 61syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) (((trL‘𝐾)‘𝑊)‘(1st𝑎)))
63 eqid 2734 . . . . . . . . . . . 12 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
6419, 20, 2, 63, 21dibelval1st 41131 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6531, 33, 34, 64syl3anc 1370 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6619, 20, 2, 10, 46, 63diatrl 41026 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6731, 33, 65, 66syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6819, 20, 45, 50, 60, 56, 62, 67lattrd 18503 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) 𝑋)
6919, 20, 2, 63, 21dibelval1st 41131 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7031, 33, 39, 69syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7119, 20, 2, 10, 46, 63diatrl 41026 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7231, 33, 70, 71syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7319, 20, 53latjle12 18507 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵 ∧ (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵𝑋𝐵)) → (((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) 𝑋 ∧ (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋) ↔ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))) 𝑋))
7445, 50, 52, 56, 73syl13anc 1371 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) 𝑋 ∧ (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋) ↔ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))) 𝑋))
7568, 72, 74mpbi2and 712 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st𝑏))) 𝑋)
7619, 20, 45, 48, 55, 56, 58, 75lattrd 18503 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) 𝑋)
7719, 20, 2, 10, 46, 63diaelval 41015 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) 𝑋)))
7877adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) 𝑋)))
7943, 76, 78mpbir2and 713 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
8030, 79eqeltrid 2842 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
81 eqid 2734 . . . . . . . . 9 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))
82 eqid 2734 . . . . . . . . 9 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
832, 10, 3, 4, 5, 81, 82dvhfplusr 41066 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
8483ad2antrr 726 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
8525, 28op2nd 8021 . . . . . . . 8 (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥 ∘ (2nd𝑎))
86 eqid 2734 . . . . . . . . . . . 12 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
8719, 20, 2, 10, 86, 21dibelval2nd 41134 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8831, 33, 34, 87syl3anc 1370 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8988coeq2d 5875 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))))
9019, 2, 10, 3, 86tendo0mulr 40809 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9131, 32, 90syl2anc 584 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9289, 91eqtrd 2774 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9385, 92eqtrid 2786 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9419, 20, 2, 10, 86, 21dibelval2nd 41134 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9531, 33, 39, 94syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9684, 93, 95oveq123d 7451 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) = (( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))))
97 simpllr 776 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑊𝐻)
9819, 2, 10, 3, 86tendo0cl 40772 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
9998ad2antrr 726 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
10019, 2, 10, 3, 86, 81tendo0pl 40773 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊)) → (( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
10144, 97, 99, 100syl21anc 838 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
10296, 101eqtrd 2774 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
103 ovex 7463 . . . . . 6 ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ V
104103elsn 4645 . . . . 5 (((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ↔ ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
105102, 104sylibr 234 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})
106 opelxpi 5725 . . . 4 ((((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) → ⟨((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)), ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏))⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
10780, 105, 106syl2anc 584 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ⟨((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)), ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏))⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
10823adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
109108, 34sseldd 3995 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
110 eqid 2734 . . . . . . 7 ( ·𝑠𝑈) = ( ·𝑠𝑈)
1112, 10, 3, 4, 110dvhvsca 41083 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
11231, 32, 109, 111syl12anc 837 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
113112oveq1d 7445 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥( ·𝑠𝑈)𝑎)(+g𝑈)𝑏) = (⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩(+g𝑈)𝑏))
11488, 99eqeltrd 2838 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊))
1152, 3tendococl 40754 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
11631, 32, 114, 115syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
117 opelxpi 5725 . . . . . 6 (((𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
11838, 116, 117syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
119108, 39sseldd 3995 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
120 eqid 2734 . . . . . 6 (+g𝑈) = (+g𝑈)
1212, 10, 3, 4, 5, 120, 82dvhvadd 41074 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ∧ 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩(+g𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)), ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏))⟩)
12231, 118, 119, 121syl12anc 837 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩(+g𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)), ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏))⟩)
123113, 122eqtrd 2774 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥( ·𝑠𝑈)𝑎)(+g𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)), ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏))⟩)
12419, 20, 2, 10, 86, 63, 21dibval2 41126 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
125124adantr 480 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
126107, 123, 1253eltr4d 2853 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥( ·𝑠𝑈)𝑎)(+g𝑈)𝑏) ∈ (𝐼𝑋))
1271, 9, 14, 15, 16, 18, 23, 24, 126islssd 20950 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wss 3962  {csn 4630  cop 4636   class class class wbr 5147  cmpt 5230   I cid 5581   × cxp 5686  cres 5690  ccom 5692  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  Basecbs 17244  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  lecple 17304  joincjn 18368  Latclat 18488  LSubSpclss 20946  HLchlt 39331  LHypclh 39966  LTrncltrn 40083  trLctrl 40140  TEndoctendo 40734  DIsoAcdia 41010  DVecHcdvh 41060  DIsoBcdib 41120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-riotaBAD 38934
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-undef 8296  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-p1 18483  df-lat 18489  df-clat 18556  df-lss 20947  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-llines 39480  df-lplanes 39481  df-lvols 39482  df-lines 39483  df-psubsp 39485  df-pmap 39486  df-padd 39778  df-lhyp 39970  df-laut 39971  df-ldil 40086  df-ltrn 40087  df-trl 40141  df-tendo 40737  df-edring 40739  df-disoa 41011  df-dvech 41061  df-dib 41121
This theorem is referenced by:  diblsmopel  41153  cdlemn5pre  41182  cdlemn11c  41191  dihjustlem  41198  dihord1  41200  dihord2a  41201  dihord2b  41202  dihord11c  41206  dihlsscpre  41216  dihopelvalcpre  41230  dihlss  41232  dihord6apre  41238  dihord5b  41241  dihord5apre  41244
  Copyright terms: Public domain W3C validator