Theorem diblss 41189

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 2736	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diblss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2735	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diblss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2735	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2735	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 41102	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2741	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
9	8	adantr 480	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
10		eqid 2735	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2735	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 41106	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2741	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	13	adantr 480	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
15		eqidd 2736	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (+g‘𝑈) = (+g‘𝑈))
16		eqidd 2736	. 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 41188	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3996	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dibn0 41172	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅)
25		fvex 6889	. . . . . . 7 ⊢ (𝑥‘(1st ‘𝑎)) ∈ V
26		vex 3463	. . . . . . . 8 ⊢ 𝑥 ∈ V
27		fvex 6889	. . . . . . . 8 ⊢ (2nd ‘𝑎) ∈ V
28	26, 27	coex 7926	. . . . . . 7 ⊢ (𝑥 ∘ (2nd ‘𝑎)) ∈ V
29	25, 28	op1st 7996	. . . . . 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 766	. . . . . . 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 1195	. . . . . . . 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 768	. . . . . . . . 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 1196	. . . . . . . . 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 41169	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊))
36	31, 33, 34, 35	syl3anc 1373	. . . . . . . 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 40786	. . . . . . . 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 1373	. . . . . . 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 1197	. . . . . . . 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 41169	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊))
41	31, 33, 39, 40	syl3anc 1373	. . . . . . 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 40738	. . . . . . 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 1373	. . . . . 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 774	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ HL)
45	44	hllatd 39382	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ Lat)
46		eqid 2735	. . . . . . . . 9 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
47	19, 2, 10, 46	trlcl 40183	. . . . . . . 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 584	. . . . . . 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 40183	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵)
50	31, 38, 49	syl2anc 584	. . . . . . . 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 40183	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵)
52	31, 41, 51	syl2anc 584	. . . . . . . 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 2735	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
54	19, 53	latjcl 18449	. . . . . . . 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 1373	. . . . . . 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 776	. . . . . . 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 40746	. . . . . . . 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 1373	. . . . . . 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 40183	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ∈ 𝐵)
60	31, 36, 59	syl2anc 584	. . . . . . . . 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 40780	. . . . . . . . . 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 1373	. . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
64	19, 20, 2, 63, 21	dibelval1st 41168	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋)) → (1st ‘𝑎) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
65	31, 33, 34, 64	syl3anc 1373	. . . . . . . . . 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 41063	. . . . . . . . . 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 1373	. . . . . . . . 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 18456	. . . . . . . 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 41168	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑏 ∈ (𝐼‘𝑋)) → (1st ‘𝑏) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
70	31, 33, 39, 69	syl3anc 1373	. . . . . . . . 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 41063	. . . . . . . . 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 1373	. . . . . . . 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 18460	. . . . . . . . 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 1374	. . . . . . . 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 712	. . . . . . 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 18456	. . . . . 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 41052	. . . . . . 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 480	. . . . . 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 713	. . . . 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 2838	. . . 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 2735	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
83	2, 10, 3, 4, 5, 81, 82	dvhfplusr 41103	. . . . . . . 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 726	. . . . . . 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 7997	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥 ∘ (2nd ‘𝑎))
86		eqid 2735	. . . . . . . . . . . 12 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
87	19, 20, 2, 10, 86, 21	dibelval2nd 41171	. . . . . . . . . . 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 1373	. . . . . . . . . 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 40846	. . . . . . . . . 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 584	. . . . . . . . 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 2770	. . . . . . . 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 2782	. . . . . . 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 41171	. . . . . . . 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 1373	. . . . . . 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 7426	. . . . . 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 775	. . . . . . 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 40809	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
99	98	ad2antrr 726	. . . . . . 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 40810	. . . . . . 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 837	. . . . . 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 2770	. . . . 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 7438	. . . . . 6 ⊢ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ V
104	103	elsn 4616	. . . . 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 234	. . . 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 5691	. . . 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 584	. . 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 480	. . . . . . 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 3959	. . . . . 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 2735	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
111	2, 10, 3, 4, 110	dvhvsca 41120	. . . . . 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 836	. . . . 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 7420	. . . 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 2834	. . . . . . 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 40791	. . . . . . 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 1373	. . . . . 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 5691	. . . . . 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 584	. . . . 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 3959	. . . . 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 2735	. . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈)
121	2, 10, 3, 4, 5, 120, 82	dvhvadd 41111	. . . . 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 836	. . . 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 2770	. . 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 41163	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
125	124	adantr 480	. . 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 2849	. 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 20892	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  {csn 4601  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   I cid 5547   × cxp 5652   ↾ cres 5656   ∘ ccom 5658  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  +_gcplusg 17271  Scalarcsca 17274   ·_𝑠 cvsca 17275  lecple 17278  joincjn 18323  Latclat 18441  LSubSpclss 20888  HLchlt 39368  LHypclh 40003  LTrncltrn 40120  trLctrl 40177  TEndoctendo 40771  DIsoAcdia 41047  DVecHcdvh 41097  DIsoBcdib 41157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-riotaBAD 38971

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-undef 8272  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-p1 18436  df-lat 18442  df-clat 18509  df-lss 20889  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39517  df-lplanes 39518  df-lvols 39519  df-lines 39520  df-psubsp 39522  df-pmap 39523  df-padd 39815  df-lhyp 40007  df-laut 40008  df-ldil 40123  df-ltrn 40124  df-trl 40178  df-tendo 40774  df-edring 40776  df-disoa 41048  df-dvech 41098  df-dib 41158

This theorem is referenced by:  diblsmopel  41190  cdlemn5pre  41219  cdlemn11c  41228  dihjustlem  41235  dihord1  41237  dihord2a  41238  dihord2b  41239  dihord11c  41243  dihlsscpre  41253  dihopelvalcpre  41267  dihlss  41269  dihord6apre  41275  dihord5b  41278  dihord5apre  41281