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Theorem diblss 35925
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 2627 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2 diblss.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 eqid 2626 . . . . 5 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4 diblss.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 eqid 2626 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2626 . . . . 5 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
72, 3, 4, 5, 6dvhbase 35838 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
87eqcomd 2632 . . 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 2626 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11 eqid 2626 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
122, 10, 3, 4, 11dvhvbase 35842 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1312eqcomd 2632 . . 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 2627 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (+g𝑈) = (+g𝑈))
16 eqidd 2627 . 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 35924 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (Base‘𝑈))
2322, 14sseqtr4d 3626 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
2419, 20, 2, 21dibn0 35908 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
25 fvex 6160 . . . . . . 7 (𝑥‘(1st𝑎)) ∈ V
26 vex 3194 . . . . . . . 8 𝑥 ∈ V
27 fvex 6160 . . . . . . . 8 (2nd𝑎) ∈ V
2826, 27coex 7068 . . . . . . 7 (𝑥 ∘ (2nd𝑎)) ∈ V
2925, 28op1st 7124 . . . . . 6 (1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥‘(1st𝑎))
3029coeq1i 5246 . . . . 5 ((1st ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) ∘ (1st𝑏)) = ((𝑥‘(1st𝑎)) ∘ (1st𝑏))
31 simpll 789 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
32 simpr1 1065 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
33 simplr 791 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑋𝐵𝑋 𝑊))
34 simpr2 1066 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (𝐼𝑋))
3519, 20, 2, 10, 21dibelval1st1 35905 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
3631, 33, 34, 35syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
372, 10, 3tendocl 35521 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
3831, 32, 36, 37syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
39 simpr3 1067 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (𝐼𝑋))
4019, 20, 2, 10, 21dibelval1st1 35905 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
4131, 33, 39, 40syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
422, 10ltrnco 35473 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
4331, 38, 41, 42syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
44 simplll 797 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝐾 ∈ HL)
45 hllat 34116 . . . . . . . 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 2626 . . . . . . . . 9 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
4819, 2, 10, 47trlcl 34917 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st𝑎)) ∘ (1st𝑏))) ∈ 𝐵)
4931, 43, 48syl2anc 692 . . . . . . 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 34917 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5131, 38, 50syl2anc 692 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st𝑎))) ∈ 𝐵)
5219, 2, 10, 47trlcl 34917 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
5331, 41, 52syl2anc 692 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) ∈ 𝐵)
54 eqid 2626 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
5519, 54latjcl 16967 . . . . . . . 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 1323 . . . . . . 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 799 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑋𝐵)
5820, 54, 2, 10, 47trlco 35481 . . . . . . . 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 1323 . . . . . . 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 34917 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6131, 36, 60syl2anc 692 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) ∈ 𝐵)
6220, 2, 10, 47, 3tendotp 35515 . . . . . . . . . 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 1323 . . . . . . . . 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 2626 . . . . . . . . . . . 12 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
6519, 20, 2, 64, 21dibelval1st 35904 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
6631, 33, 34, 65syl3anc 1323 . . . . . . . . . 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 35799 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑎)) 𝑋)
6831, 33, 66, 67syl3anc 1323 . . . . . . . . 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 16974 . . . . . . . 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 35904 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7131, 33, 39, 70syl3anc 1323 . . . . . . . . 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 35799 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7331, 33, 71, 72syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st𝑏)) 𝑋)
7419, 20, 54latjle12 16978 . . . . . . . . 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 1325 . . . . . . . 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 955 . . . . . . 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 16974 . . . . . 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 35788 . . . . . . 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 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)))
8043, 77, 79mpbir2and 956 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → ((𝑥‘(1st𝑎)) ∘ (1st𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
8130, 80syl5eqel 2708 . . . 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 2626 . . . . . . . . 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 2626 . . . . . . . . 9 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
842, 10, 3, 4, 5, 82, 83dvhfplusr 35839 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
8584ad2antrr 761 . . . . . . 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 7125 . . . . . . . 8 (2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩) = (𝑥 ∘ (2nd𝑎))
87 eqid 2626 . . . . . . . . . . . 12 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
8819, 20, 2, 10, 87, 21dibelval2nd 35907 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑎 ∈ (𝐼𝑋)) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
8931, 33, 34, 88syl3anc 1323 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9089coeq2d 5249 . . . . . . . . 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 35581 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9231, 32, 91syl2anc 692 . . . . . . . . 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 2660 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9486, 93syl5eq 2672 . . . . . . 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 35907 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑏 ∈ (𝐼𝑋)) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9631, 33, 39, 95syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑏) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
9785, 94, 96oveq123d 6626 . . . . . 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 798 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑊𝐻)
9919, 2, 10, 3, 87tendo0cl 35544 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
10099ad2antrr 761 . . . . . . 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 35545 . . . . . . 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 1322 . . . . . 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 2660 . . . . 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 6633 . . . . . 6 ((2nd ‘⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd𝑏)) ∈ V
105104elsn 4168 . . . . 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 224 . . . 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 5113 . . . 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 692 . . 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 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝐼𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
110109, 34sseldd 3589 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
111 eqid 2626 . . . . . . 7 ( ·𝑠𝑈) = ( ·𝑠𝑈)
1122, 10, 3, 4, 111dvhvsca 35856 . . . . . 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 1321 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥( ·𝑠𝑈)𝑎) = ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩)
114113oveq1d 6620 . . . 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 2704 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊))
1162, 3tendococl 35526 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑎) ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
11731, 32, 115, 116syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
118 opelxpi 5113 . . . . . 6 (((𝑥‘(1st𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑥 ∘ (2nd𝑎)) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨(𝑥‘(1st𝑎)), (𝑥 ∘ (2nd𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
11938, 117, 118syl2anc 692 . . . . 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 3589 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼𝑋) ∧ 𝑏 ∈ (𝐼𝑋))) → 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
121 eqid 2626 . . . . . 6 (+g𝑈) = (+g𝑈)
1222, 10, 3, 4, 5, 121, 83dvhvadd 35847 . . . . 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 1321 . . . 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 2660 . . 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 35899 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
126125adantr 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))}))
127108, 124, 1263eltr4d 2719 . 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 18850 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wss 3560  {csn 4153  cop 4159   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  cres 5081  ccom 5083  cfv 5850  (class class class)co 6605  cmpt2 6607  1st c1st 7114  2nd c2nd 7115  Basecbs 15776  +gcplusg 15857  Scalarcsca 15860   ·𝑠 cvsca 15861  lecple 15864  joincjn 16860  Latclat 16961  LSubSpclss 18846  HLchlt 34103  LHypclh 34736  LTrncltrn 34853  trLctrl 34911  TEndoctendo 35506  DIsoAcdia 35783  DVecHcdvh 35833  DIsoBcdib 35893
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-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-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  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-lss 18847  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-disoa 35784  df-dvech 35834  df-dib 35894
This theorem is referenced by:  diblsmopel  35926  cdlemn5pre  35955  cdlemn11c  35964  dihjustlem  35971  dihord1  35973  dihord2a  35974  dihord2b  35975  dihord11c  35979  dihlsscpre  35989  dihopelvalcpre  36003  dihlss  36005  dihord6apre  36011  dihord5b  36014  dihord5apre  36017
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