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Theorem diblss 35477
 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 2611 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2 diblss.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 eqid 2610 . . . . 5 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4 diblss.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 eqid 2610 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2610 . . . . 5 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
72, 3, 4, 5, 6dvhbase 35390 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
87eqcomd 2616 . . 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 2610 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11 eqid 2610 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
122, 10, 3, 4, 11dvhvbase 35394 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1312eqcomd 2616 . . 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 2611 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (+g𝑈) = (+g𝑈))
16 eqidd 2611 . 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 35476 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (Base‘𝑈))
2322, 14sseqtr4d 3605 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
2419, 20, 2, 21dibn0 35460 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
25 fvex 6113 . . . . . . 7 (𝑥‘(1st𝑎)) ∈ V
26 vex 3176 . . . . . . . 8 𝑥 ∈ V
27 fvex 6113 . . . . . . . 8 (2nd𝑎) ∈ V
2826, 27coex 7011 . . . . . . 7 (𝑥 ∘ (2nd𝑎)) ∈ V
2925, 28op1st 7067 . . . . . 6 (1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥‘(1st𝑎))
3029coeq1i 5203 . . . . 5 ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) = ((𝑥‘(1st𝑎)) ∘ (1st𝑏))
31 simpll 786 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
32 simpr1 1060 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
33 simplr 788 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑋𝐵𝑋 𝑊))
34 simpr2 1061 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (𝐼𝑋))
3519, 20, 2, 10, 21dibelval1st1 35457 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
3631, 33, 34, 35syl3anc 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
372, 10, 3tendocl 35073 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
3831, 32, 36, 37syl3anc 1318 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
39 simpr3 1062 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (𝐼𝑋))
4019, 20, 2, 10, 21dibelval1st1 35457 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
4131, 33, 39, 40syl3anc 1318 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
422, 10ltrnco 35025 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
4331, 38, 41, 42syl3anc 1318 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
44 simplll 794 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ HL)
45 hllat 33668 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4644, 45syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ Lat)
47 eqid 2610 . . . . . . . . 9 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
4819, 2, 10, 47trlcl 34469 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ∈ 𝐵)
4931, 43, 48syl2anc 691 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ∈ 𝐵)
5019, 2, 10, 47trlcl 34469 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5131, 38, 50syl2anc 691 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5219, 2, 10, 47trlcl 34469 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
5331, 41, 52syl2anc 691 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
54 eqid 2610 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
5519, 54latjcl 16874 . . . . . . . 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)
5646, 51, 53, 55syl3anc 1318 . . . . . . 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)
57 simplrl 796 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑋𝐵)
5820, 54, 2, 10, 47trlco 35033 . . . . . . . 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))))
5931, 38, 41, 58syl3anc 1318 . . . . . . 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))))
6019, 2, 10, 47trlcl 34469 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6131, 36, 60syl2anc 691 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6220, 2, 10, 47, 3tendotp 35067 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) (((trL‘𝐾)‘𝑊)‘(1st𝑎)))
6331, 32, 36, 62syl3anc 1318 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) (((trL‘𝐾)‘𝑊)‘(1st𝑎)))
64 eqid 2610 . . . . . . . . . . . 12 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
6519, 20, 2, 64, 21dibelval1st 35456 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6631, 33, 34, 65syl3anc 1318 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6719, 20, 2, 10, 47, 64diatrl 35351 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6831, 33, 66, 67syl3anc 1318 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6919, 20, 46, 51, 61, 57, 63, 68lattrd 16881 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) 𝑋)
7019, 20, 2, 64, 21dibelval1st 35456 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7131, 33, 39, 70syl3anc 1318 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7219, 20, 2, 10, 47, 64diatrl 35351 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7331, 33, 71, 72syl3anc 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7419, 20, 54latjle12 16885 . . . . . . . . 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))
7546, 51, 53, 57, 74syl13anc 1320 . . . . . . . 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))
7669, 73, 75mpbi2and 958 . . . . . . 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)
7719, 20, 46, 49, 56, 57, 59, 76lattrd 16881 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) 𝑋)
7819, 20, 2, 10, 47, 64diaelval 35340 . . . . . . 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)))
7978adantr 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)))
8043, 77, 79mpbir2and 959 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
8130, 80syl5eqel 2692 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
82 eqid 2610 . . . . . . . . 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))))
83 eqid 2610 . . . . . . . . 9 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
842, 10, 3, 4, 5, 82, 83dvhfplusr 35391 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
8584ad2antrr 758 . . . . . . 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)))))
8625, 28op2nd 7068 . . . . . . . 8 (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥 ∘ (2nd𝑎))
87 eqid 2610 . . . . . . . . . . . 12 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
8819, 20, 2, 10, 87, 21dibelval2nd 35459 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8931, 33, 34, 88syl3anc 1318 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9089coeq2d 5206 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))))
9119, 2, 10, 3, 87tendo0mulr 35133 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9231, 32, 91syl2anc 691 . . . . . . . . 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)))
9390, 92eqtrd 2644 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9486, 93syl5eq 2656 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9519, 20, 2, 10, 87, 21dibelval2nd 35459 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9631, 33, 39, 95syl3anc 1318 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9785, 94, 96oveq123d 6570 . . . . . 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))))
98 simpllr 795 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑊𝐻)
9919, 2, 10, 3, 87tendo0cl 35096 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
10099ad2antrr 758 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
10119, 2, 10, 3, 87, 82tendo0pl 35097 . . . . . . 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)))
10244, 98, 100, 101syl21anc 1317 . . . . . 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)))
10397, 102eqtrd 2644 . . . . 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)))
104 ovex 6577 . . . . . 6 ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ V
105104elsn 4140 . . . . 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)))
106103, 105sylibr 223 . . . 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))})
107 opelxpi 5072 . . . 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))}))
10881, 106, 107syl2anc 691 . . 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))}))
10923adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
110109, 34sseldd 3569 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
111 eqid 2610 . . . . . . 7 ( ·𝑠𝑈) = ( ·𝑠𝑈)
1122, 10, 3, 4, 111dvhvsca 35408 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
11331, 32, 110, 112syl12anc 1316 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
114113oveq1d 6564 . . . 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))
11589, 100eqeltrd 2688 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊))
1162, 3tendococl 35078 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
11731, 32, 115, 116syl3anc 1318 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
118 opelxpi 5072 . . . . . 6 (((𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
11938, 117, 118syl2anc 691 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
120109, 39sseldd 3569 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
121 eqid 2610 . . . . . 6 (+g𝑈) = (+g𝑈)
1222, 10, 3, 4, 5, 121, 83dvhvadd 35399 . . . . 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))⟩)
12331, 119, 120, 122syl12anc 1316 . . . 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))⟩)
124114, 123eqtrd 2644 . . 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))⟩)
12519, 20, 2, 10, 87, 64, 21dibval2 35451 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
126125adantr 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))}))
127108, 124, 1263eltr4d 2703 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥( ·𝑠𝑈)𝑎)(+g𝑈)𝑏) ∈ (𝐼𝑋))
1281, 9, 14, 15, 16, 18, 23, 24, 127islssd 18757 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  lecple 15775  joincjn 16767  Latclat 16868  LSubSpclss 18753  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  TEndoctendo 35058  DIsoAcdia 35335  DVecHcdvh 35385  DIsoBcdib 35445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-undef 7286  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-lss 18754  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tendo 35061  df-edring 35063  df-disoa 35336  df-dvech 35386  df-dib 35446 This theorem is referenced by:  diblsmopel  35478  cdlemn5pre  35507  cdlemn11c  35516  dihjustlem  35523  dihord1  35525  dihord2a  35526  dihord2b  35527  dihord11c  35531  dihlsscpre  35541  dihopelvalcpre  35555  dihlss  35557  dihord6apre  35563  dihord5b  35566  dihord5apre  35569
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