Theorem diblss 42493

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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Proof of Theorem diblss

Dummy variables 𝑎 𝑏 𝑥 ℎ 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2764	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diblss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2763	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diblss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2763	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2763	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 42406	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2769	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
9	8	adantr 486	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
10		eqid 2763	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2763	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 42410	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2769	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	13	adantr 486	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
15		eqidd 2764	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (+g‘𝑈) = (+g‘𝑈))
16		eqidd 2764	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈))
17		diblss.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
18	17	a1i 11	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 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)
22	19, 20, 2, 21, 4, 11	dibss 42492	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3950	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dibn0 42476	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅)
25		fvex 6929	. . . . . . 7 ⊢ (𝑥‘(1st ‘𝑎)) ∈ V
26		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
27		fvex 6929	. . . . . . . 8 ⊢ (2nd ‘𝑎) ∈ V
28	26, 27	coex 7976	. . . . . . 7 ⊢ (𝑥 ∘ (2nd ‘𝑎)) ∈ V
29	25, 28	op1st 8043	. . . . . 6 ⊢ (1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥‘(1st ‘𝑎))
30	29	coeq1i 5848	. . . . 5 ⊢ ((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)) = ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))
31		simpll 779	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32		simpr1 1213	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
33		simplr 781	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
34		simpr2 1214	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑎 ∈ (𝐼‘𝑋))
35	19, 20, 2, 10, 21	dibelval1st1 42473	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
36	31, 33, 34, 35	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
37	2, 10, 3	tendocl 42090	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
38	31, 32, 36, 37	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊))
39		simpr3 1215	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑏 ∈ (𝐼‘𝑋))
40	19, 20, 2, 10, 21	dibelval1st1 42473	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
41	31, 33, 39, 40	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
42	2, 10	ltrnco 42042	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
43	31, 38, 41, 42	syl3anc 1399	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ ((LTrn‘𝐾)‘𝑊))
44		simplll 787	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ HL)
45	44	hllatd 40687	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ Lat)
46		eqid 2763	. . . . . . . . 9 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
47	19, 2, 10, 46	trlcl 41487	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ∈ 𝐵)
48	31, 43, 47	syl2anc 596	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ∈ 𝐵)
49	19, 2, 10, 46	trlcl 41487	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵)
50	31, 38, 49	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵)
51	19, 2, 10, 46	trlcl 41487	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵)
52	31, 41, 51	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵)
53		eqid 2763	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
54	19, 53	latjcl 18663	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵 ∧ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))) ∈ 𝐵)
55	45, 50, 52, 54	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))) ∈ 𝐵)
56		simplrl 789	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑋 ∈ 𝐵)
57	20, 53, 2, 10, 46	trlco 42050	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ≤ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))))
58	31, 38, 41, 57	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ≤ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))))
59	19, 2, 10, 46	trlcl 41487	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ∈ 𝐵)
60	31, 36, 59	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ∈ 𝐵)
61	20, 2, 10, 46, 3	tendotp 42084	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ≤ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)))
62	31, 32, 36, 61	syl3anc 1399	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ≤ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)))
63		eqid 2763	. . . . . . . . . . . 12 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
64	19, 20, 2, 63, 21	dibelval1st 42472	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
65	31, 33, 34, 64	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (1st ‘𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
66	19, 20, 2, 10, 46, 63	diatrl 42367	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ≤ 𝑋)
67	31, 33, 65, 66	syl3anc 1399	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ≤ 𝑋)
68	19, 20, 45, 50, 60, 56, 62, 67	lattrd 18670	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ≤ 𝑋)
69	19, 20, 2, 63, 21	dibelval1st 42472	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
70	31, 33, 39, 69	syl3anc 1399	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (1st ‘𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
71	19, 20, 2, 10, 46, 63	diatrl 42367	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ≤ 𝑋)
72	31, 33, 70, 71	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ≤ 𝑋)
73	19, 20, 53	latjle12 18674	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵 ∧ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ≤ 𝑋 ∧ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ≤ 𝑋) ↔ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))) ≤ 𝑋))
74	45, 50, 52, 56, 73	syl13anc 1400	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ≤ 𝑋 ∧ (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ≤ 𝑋) ↔ ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))) ≤ 𝑋))
75	68, 72, 74	mpbi2and 725	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎)))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(1st ‘𝑏))) ≤ 𝑋)
76	19, 20, 45, 48, 55, 56, 58, 75	lattrd 18670	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ≤ 𝑋)
77	19, 20, 2, 10, 46, 63	diaelval 42356	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ≤ 𝑋)))
78	77	adantr 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (((trL‘𝐾)‘𝑊)‘((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))) ≤ 𝑋)))
79	43, 76, 78	mpbir2and 726	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
80	30, 79	eqeltrid 2868	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
81		eqid 2763	. . . . . . . . 9 ⊢ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))
82		eqid 2763	. . . . . . . . 9 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
83	2, 10, 3, 4, 5, 81, 82	dvhfplusr 42407	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ)))))
84	83	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ)))))
85	25, 28	op2nd 8044	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥 ∘ (2nd ‘𝑎))
86		eqid 2763	. . . . . . . . . . . 12 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
87	19, 20, 2, 10, 86, 21	dibelval2nd 42475	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (2nd ‘𝑎) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
88	31, 33, 34, 87	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (2nd ‘𝑎) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
89	88	coeq2d 5851	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥 ∘ (2nd ‘𝑎)) = (𝑥 ∘ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))))
90	19, 2, 10, 3, 86	tendo0mulr 42150	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
91	31, 32, 90	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥 ∘ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
92	89, 91	eqtrd 2798	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥 ∘ (2nd ‘𝑎)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
93	85, 92	eqtrid 2810	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
94	19, 20, 2, 10, 86, 21	dibelval2nd 42475	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (2nd ‘𝑏) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
95	31, 33, 39, 94	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (2nd ‘𝑏) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
96	84, 93, 95	oveq123d 7472	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) = ((ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))))
97		simpllr 788	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑊 ∈ 𝐻)
98	19, 2, 10, 3, 86	tendo0cl 42113	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
99	98	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
100	19, 2, 10, 3, 86, 81	tendo0pl 42114	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊)) → ((ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
101	44, 97, 99, 100	syl21anc 851	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
102	96, 101	eqtrd 2798	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
103		ovex 7484	. . . . . 6 ⊢ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ V
104	103	elsn 4599	. . . . 5 ⊢ (((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))} ↔ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)))
105	102, 104	sylibr 237	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})
106		opelxpi 5695	. . . 4 ⊢ ((((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) → ⟨((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)), ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏))⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
107	80, 105, 106	syl2anc 596	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ⟨((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)), ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏))⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
108	23	adantr 486	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
109	108, 34	sseldd 3914	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
110		eqid 2763	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
111	2, 10, 3, 4, 110	dvhvsca 42424	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑥( ·𝑠 ‘𝑈)𝑎) = ⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)
112	31, 32, 109, 111	syl12anc 850	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥( ·𝑠 ‘𝑈)𝑎) = ⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)
113	112	oveq1d 7466	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((𝑥( ·𝑠 ‘𝑈)𝑎)(+g‘𝑈)𝑏) = (⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩(+g‘𝑈)𝑏))
114	88, 99	eqeltrd 2864	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (2nd ‘𝑎) ∈ ((TEndo‘𝐾)‘𝑊))
115	2, 3	tendococl 42095	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd ‘𝑎) ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ (2nd ‘𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
116	31, 32, 114, 115	syl3anc 1399	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝑥 ∘ (2nd ‘𝑎)) ∈ ((TEndo‘𝐾)‘𝑊))
117		opelxpi 5695	. . . . . 6 ⊢ (((𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑥 ∘ (2nd ‘𝑎)) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
118	38, 116, 117	syl2anc 596	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
119	108, 39	sseldd 3914	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
120		eqid 2763	. . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈)
121	2, 10, 3, 4, 5, 120, 82	dvhvadd 42415	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩ ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ∧ 𝑏 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩(+g‘𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)), ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏))⟩)
122	31, 118, 119, 121	syl12anc 850	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩(+g‘𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)), ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏))⟩)
123	113, 122	eqtrd 2798	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((𝑥( ·𝑠 ‘𝑈)𝑎)(+g‘𝑈)𝑏) = ⟨((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)), ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏))⟩)
124	19, 20, 2, 10, 86, 63, 21	dibval2 42467	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
125	124	adantr 486	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
126	107, 123, 125	3eltr4d 2879	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → ((𝑥( ·𝑠 ‘𝑈)𝑎)(+g‘𝑈)𝑏) ∈ (𝐼‘𝑋))
127	1, 9, 14, 15, 16, 18, 23, 24, 126	islssd 21261	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  {csn 4588  ⟨cop 4840   class class class wbr 5108   ↦ cmpt 5192   I cid 5552   × cxp 5656   ↾ cres 5660   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431  +_gcplusg 17472  Scalarcsca 17475   ·_𝑠 cvsca 17476  lecple 17479  joincjn 18535  Latclat 18655  LSubSpclss 21257  HLchlt 40673  LHypclh 41307  LTrncltrn 41424  trLctrl 41481  TEndoctendo 42075  DIsoAcdia 42351  DVecHcdvh 42401  DIsoBcdib 42461

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-riotaBAD 40276

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-undef 8328  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-plusg 17486  df-mulr 17487  df-sca 17489  df-vsca 17490  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-p1 18648  df-lat 18656  df-clat 18723  df-lss 21258  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822  df-lvols 40823  df-lines 40824  df-psubsp 40826  df-pmap 40827  df-padd 41119  df-lhyp 41311  df-laut 41312  df-ldil 41427  df-ltrn 41428  df-trl 41482  df-tendo 42078  df-edring 42080  df-disoa 42352  df-dvech 42402  df-dib 42462

This theorem is used by:  diblsmopel  42494  cdlemn5pre  42523  cdlemn11c  42532  dihjustlem  42539  dihord1  42541  dihord2a  42542  dihord2b  42543  dihord11c  42547  dihlsscpre  42557  dihopelvalcpre  42571  dihlss  42573  dihord6apre  42579  dihord5b  42582  dihord5apre  42585