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 38175
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 2826 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2 diblss.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 eqid 2825 . . . . 5 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4 diblss.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 eqid 2825 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2825 . . . . 5 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
72, 3, 4, 5, 6dvhbase 38088 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
87eqcomd 2831 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
98adantr 481 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
10 eqid 2825 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11 eqid 2825 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
122, 10, 3, 4, 11dvhvbase 38092 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1312eqcomd 2831 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
1413adantr 481 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
15 eqidd 2826 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (+g𝑈) = (+g𝑈))
16 eqidd 2826 . 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 38174 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (Base‘𝑈))
2322, 14sseqtrrd 4011 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
2419, 20, 2, 21dibn0 38158 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
25 fvex 6679 . . . . . . 7 (𝑥‘(1st𝑎)) ∈ V
26 vex 3502 . . . . . . . 8 𝑥 ∈ V
27 fvex 6679 . . . . . . . 8 (2nd𝑎) ∈ V
2826, 27coex 7626 . . . . . . 7 (𝑥 ∘ (2nd𝑎)) ∈ V
2925, 28op1st 7691 . . . . . 6 (1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥‘(1st𝑎))
3029coeq1i 5728 . . . . 5 ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) = ((𝑥‘(1st𝑎)) ∘ (1st𝑏))
31 simpll 763 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
32 simpr1 1188 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
33 simplr 765 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑋𝐵𝑋 𝑊))
34 simpr2 1189 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (𝐼𝑋))
3519, 20, 2, 10, 21dibelval1st1 38155 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
3631, 33, 34, 35syl3anc 1365 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
372, 10, 3tendocl 37772 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
3831, 32, 36, 37syl3anc 1365 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
39 simpr3 1190 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (𝐼𝑋))
4019, 20, 2, 10, 21dibelval1st1 38155 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
4131, 33, 39, 40syl3anc 1365 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
422, 10ltrnco 37724 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
4331, 38, 41, 42syl3anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
44 simplll 771 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ HL)
4544hllatd 36369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ Lat)
46 eqid 2825 . . . . . . . . 9 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
4719, 2, 10, 46trlcl 37169 . . . . . . . 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 37169 . . . . . . . . 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 37169 . . . . . . . . 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 2825 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
5419, 53latjcl 17653 . . . . . . . 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 1365 . . . . . . 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 773 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑋𝐵)
5720, 53, 2, 10, 46trlco 37732 . . . . . . . 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 1365 . . . . . . 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 37169 . . . . . . . . . 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 37766 . . . . . . . . . 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 1365 . . . . . . . . 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 2825 . . . . . . . . . . . 12 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
6419, 20, 2, 63, 21dibelval1st 38154 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6531, 33, 34, 64syl3anc 1365 . . . . . . . . . 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 38049 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6731, 33, 65, 66syl3anc 1365 . . . . . . . . 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 17660 . . . . . . . 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 38154 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7031, 33, 39, 69syl3anc 1365 . . . . . . . . 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 38049 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7231, 33, 70, 71syl3anc 1365 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7319, 20, 53latjle12 17664 . . . . . . . . 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 1366 . . . . . . . 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 708 . . . . . . 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 17660 . . . . . 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 38038 . . . . . . 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 481 . . . . . 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 709 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
8030, 79eqeltrid 2921 . . . 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 2825 . . . . . . . . 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 2825 . . . . . . . . 9 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
832, 10, 3, 4, 5, 81, 82dvhfplusr 38089 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
8483ad2antrr 722 . . . . . . 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 7692 . . . . . . . 8 (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥 ∘ (2nd𝑎))
86 eqid 2825 . . . . . . . . . . . 12 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
8719, 20, 2, 10, 86, 21dibelval2nd 38157 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8831, 33, 34, 87syl3anc 1365 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8988coeq2d 5731 . . . . . . . . 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 37832 . . . . . . . . . 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 2860 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9385, 92syl5eq 2872 . . . . . . 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 38157 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9531, 33, 39, 94syl3anc 1365 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9684, 93, 95oveq123d 7172 . . . . . 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 772 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑊𝐻)
9819, 2, 10, 3, 86tendo0cl 37795 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
9998ad2antrr 722 . . . . . . 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 37796 . . . . . . 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 835 . . . . . 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 2860 . . . . 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 7184 . . . . . 6 ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ V
104103elsn 4578 . . . . 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 235 . . . 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 5590 . . . 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 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
109108, 34sseldd 3971 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
110 eqid 2825 . . . . . . 7 ( ·𝑠𝑈) = ( ·𝑠𝑈)
1112, 10, 3, 4, 110dvhvsca 38106 . . . . . 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 834 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
113112oveq1d 7166 . . . 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 2917 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊))
1152, 3tendococl 37777 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
11631, 32, 114, 115syl3anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
117 opelxpi 5590 . . . . . 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 3971 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
120 eqid 2825 . . . . . 6 (+g𝑈) = (+g𝑈)
1212, 10, 3, 4, 5, 120, 82dvhvadd 38097 . . . . 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 834 . . . 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 2860 . . 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 38149 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
125124adantr 481 . . 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 2932 . 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 19629 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3939  {csn 4563  cop 4569   class class class wbr 5062  cmpt 5142   I cid 5457   × cxp 5551  cres 5555  ccom 5557  cfv 6351  (class class class)co 7151  cmpo 7153  1st c1st 7681  2nd c2nd 7682  Basecbs 16475  +gcplusg 16557  Scalarcsca 16560   ·𝑠 cvsca 16561  lecple 16564  joincjn 17546  Latclat 17647  LSubSpclss 19625  HLchlt 36355  LHypclh 36989  LTrncltrn 37106  trLctrl 37163  TEndoctendo 37757  DIsoAcdia 38033  DVecHcdvh 38083  DIsoBcdib 38143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-riotaBAD 35958
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-undef 7933  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-lss 19626  df-oposet 36181  df-ol 36183  df-oml 36184  df-covers 36271  df-ats 36272  df-atl 36303  df-cvlat 36327  df-hlat 36356  df-llines 36503  df-lplanes 36504  df-lvols 36505  df-lines 36506  df-psubsp 36508  df-pmap 36509  df-padd 36801  df-lhyp 36993  df-laut 36994  df-ldil 37109  df-ltrn 37110  df-trl 37164  df-tendo 37760  df-edring 37762  df-disoa 38034  df-dvech 38084  df-dib 38144
This theorem is referenced by:  diblsmopel  38176  cdlemn5pre  38205  cdlemn11c  38214  dihjustlem  38221  dihord1  38223  dihord2a  38224  dihord2b  38225  dihord11c  38229  dihlsscpre  38239  dihopelvalcpre  38253  dihlss  38255  dihord6apre  38261  dihord5b  38264  dihord5apre  38267
  Copyright terms: Public domain W3C validator