Theorem diblss 42043

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 2761	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diblss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2760	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diblss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2760	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2760	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 41956	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2766	. . 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 2760	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2760	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 41960	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2766	. . 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 2761	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (+g‘𝑈) = (+g‘𝑈))
16		eqidd 2761	. 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 42042	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3968	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dibn0 42026	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅)
25		fvex 6891	. . . . . . 7 ⊢ (𝑥‘(1st ‘𝑎)) ∈ V
26		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
27		fvex 6891	. . . . . . . 8 ⊢ (2nd ‘𝑎) ∈ V
28	26, 27	coex 7927	. . . . . . 7 ⊢ (𝑥 ∘ (2nd ‘𝑎)) ∈ V
29	25, 28	op1st 7994	. . . . . 6 ⊢ (1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥‘(1st ‘𝑎))
30	29	coeq1i 5839	. . . . 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 42023	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
36	31, 33, 34, 35	syl3anc 1398	. . . . . . . 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 41640	. . . . . . . 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 1398	. . . . . . 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 42023	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
41	31, 33, 39, 40	syl3anc 1398	. . . . . . 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 41592	. . . . . . 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 1398	. . . . . 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 40237	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ Lat)
46		eqid 2760	. . . . . . . . 9 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
47	19, 2, 10, 46	trlcl 41037	. . . . . . . 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 41037	. . . . . . . . 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 41037	. . . . . . . . 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 2760	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
54	19, 53	latjcl 18527	. . . . . . . 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 1398	. . . . . . 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 41600	. . . . . . . 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 1398	. . . . . . 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 41037	. . . . . . . . . 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 41634	. . . . . . . . . 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 1398	. . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
64	19, 20, 2, 63, 21	dibelval1st 42022	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
65	31, 33, 34, 64	syl3anc 1398	. . . . . . . . . 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 41917	. . . . . . . . . 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 1398	. . . . . . . . 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 18534	. . . . . . . 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 42022	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
70	31, 33, 39, 69	syl3anc 1398	. . . . . . . . 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 41917	. . . . . . . . 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 1398	. . . . . . . 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 18538	. . . . . . . . 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 1399	. . . . . . . 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 18534	. . . . . 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 41906	. . . . . . 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 2864	. . . 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 2760	. . . . . . . . 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 2760	. . . . . . . . 9 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
83	2, 10, 3, 4, 5, 81, 82	dvhfplusr 41957	. . . . . . . 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 7995	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥 ∘ (2nd ‘𝑎))
86		eqid 2760	. . . . . . . . . . . 12 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
87	19, 20, 2, 10, 86, 21	dibelval2nd 42025	. . . . . . . . . . 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 1398	. . . . . . . . . 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 5842	. . . . . . . . 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 41700	. . . . . . . . . 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 2795	. . . . . . . 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 2807	. . . . . . 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 42025	. . . . . . . 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 1398	. . . . . . 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 7434	. . . . . 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 41663	. . . . . . . 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 41664	. . . . . . 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 2795	. . . . 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 7446	. . . . . 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 5692	. . . 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 3932	. . . . . 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 2760	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
111	2, 10, 3, 4, 110	dvhvsca 41974	. . . . . 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 7428	. . . 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 2860	. . . . . . 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 41645	. . . . . . 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 1398	. . . . . 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 5692	. . . . . 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 3932	. . . . 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 2760	. . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈)
121	2, 10, 3, 4, 5, 120, 82	dvhvadd 41965	. . . . 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 2795	. . 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 42017	. . . 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 2875	. 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 21119	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 2145   ⊆ wss 3899  {csn 4584  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   I cid 5549   × cxp 5653   ↾ cres 5657   ∘ ccom 5659  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17301  +_gcplusg 17342  Scalarcsca 17345   ·_𝑠 cvsca 17346  lecple 17349  joincjn 18399  Latclat 18519  LSubSpclss 21115  HLchlt 40223  LHypclh 40857  LTrncltrn 40974  trLctrl 41031  TEndoctendo 41625  DIsoAcdia 41901  DVecHcdvh 41951  DIsoBcdib 42011

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-riotaBAD 39826

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-undef 8271  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-proset 18382  df-poset 18401  df-plt 18416  df-lub 18432  df-glb 18433  df-join 18434  df-meet 18435  df-p0 18511  df-p1 18512  df-lat 18520  df-clat 18587  df-lss 21116  df-oposet 40049  df-ol 40051  df-oml 40052  df-covers 40139  df-ats 40140  df-atl 40171  df-cvlat 40195  df-hlat 40224  df-llines 40371  df-lplanes 40372  df-lvols 40373  df-lines 40374  df-psubsp 40376  df-pmap 40377  df-padd 40669  df-lhyp 40861  df-laut 40862  df-ldil 40977  df-ltrn 40978  df-trl 41032  df-tendo 41628  df-edring 41630  df-disoa 41902  df-dvech 41952  df-dib 42012

This theorem is used by:  diblsmopel  42044  cdlemn5pre  42073  cdlemn11c  42082  dihjustlem  42089  dihord1  42091  dihord2a  42092  dihord2b  42093  dihord11c  42097  dihlsscpre  42107  dihopelvalcpre  42121  dihlss  42123  dihord6apre  42129  dihord5b  42132  dihord5apre  42135