Theorem diblss 41944

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 41857	. . . 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 485	. 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 41861	. . . 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 485	. 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 41943	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3974	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dibn0 41927	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅)
25		fvex 6894	. . . . . . 7 ⊢ (𝑥‘(1st ‘𝑎)) ∈ V
26		vex 3459	. . . . . . . 8 ⊢ 𝑥 ∈ V
27		fvex 6894	. . . . . . . 8 ⊢ (2nd ‘𝑎) ∈ V
28	26, 27	coex 7923	. . . . . . 7 ⊢ (𝑥 ∘ (2nd ‘𝑎)) ∈ V
29	25, 28	op1st 7990	. . . . . 6 ⊢ (1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥‘(1st ‘𝑎))
30	29	coeq1i 5845	. . . . 5 ⊢ ((1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) ∘ (1st ‘𝑏)) = ((𝑥‘(1st ‘𝑎)) ∘ (1st ‘𝑏))
31		simpll 778	. . . . . . 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 780	. . . . . . . . 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 41924	. . . . . . . . 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 41541	. . . . . . . 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 41924	. . . . . . . 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 41493	. . . . . . 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 786	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ HL)
45	44	hllatd 40138	. . . . . . 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 40938	. . . . . . . 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 595	. . . . . . 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 40938	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥‘(1st ‘𝑎)) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑥‘(1st ‘𝑎))) ∈ 𝐵)
50	31, 38, 49	syl2anc 595	. . . . . . . 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 40938	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑏) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑏)) ∈ 𝐵)
52	31, 41, 51	syl2anc 595	. . . . . . . 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 18490	. . . . . . . 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 788	. . . . . . 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 41501	. . . . . . . 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 40938	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑎) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(1st ‘𝑎)) ∈ 𝐵)
60	31, 36, 59	syl2anc 595	. . . . . . . . 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 41535	. . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
64	19, 20, 2, 63, 21	dibelval1st 41923	. . . . . . . . . . 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 41818	. . . . . . . . . 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 18497	. . . . . . . 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 41923	. . . . . . . . . 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 41818	. . . . . . . . 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 18501	. . . . . . . . 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 724	. . . . . . 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 18497	. . . . . 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 41807	. . . . . . 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 485	. . . . . 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 725	. . . . 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 2867	. . . 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 41858	. . . . . . . 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 738	. . . . . . 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 7991	. . . . . . . 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 41926	. . . . . . . . . . 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 5848	. . . . . . . . 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 41601	. . . . . . . . . 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 595	. . . . . . . . 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 41926	. . . . . . . 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 7431	. . . . . 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 787	. . . . . . 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 41564	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
99	98	ad2antrr 738	. . . . . . 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 41565	. . . . . . 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 850	. . . . . 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 7443	. . . . . 6 ⊢ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ V
104	103	elsn 4604	. . . . 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 5698	. . . 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 595	. . 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 485	. . . . . . 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 3938	. . . . . 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 41875	. . . . . 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 849	. . . . 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 7425	. . . 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 2863	. . . . . . 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 41546	. . . . . . 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 5698	. . . . . 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 595	. . . . 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 3938	. . . . 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 41866	. . . . 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 849	. . . 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 41918	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {(ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
125	124	adantr 485	. . 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 2878	. 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 21056	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   I cid 5555   × cxp 5659   ↾ cres 5663   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17264  +_gcplusg 17305  Scalarcsca 17308   ·_𝑠 cvsca 17309  lecple 17312  joincjn 18362  Latclat 18482  LSubSpclss 21052  HLchlt 40124  LHypclh 40758  LTrncltrn 40875  trLctrl 40932  TEndoctendo 41526  DIsoAcdia 41802  DVecHcdvh 41852  DIsoBcdib 41912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-riotaBAD 39727

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-undef 8265  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-mulr 17319  df-sca 17321  df-vsca 17322  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-p1 18475  df-lat 18483  df-clat 18550  df-lss 21053  df-oposet 39950  df-ol 39952  df-oml 39953  df-covers 40040  df-ats 40041  df-atl 40072  df-cvlat 40096  df-hlat 40125  df-llines 40272  df-lplanes 40273  df-lvols 40274  df-lines 40275  df-psubsp 40277  df-pmap 40278  df-padd 40570  df-lhyp 40762  df-laut 40763  df-ldil 40878  df-ltrn 40879  df-trl 40933  df-tendo 41529  df-edring 41531  df-disoa 41803  df-dvech 41853  df-dib 41913

This theorem is referenced by:  diblsmopel  41945  cdlemn5pre  41974  cdlemn11c  41983  dihjustlem  41990  dihord1  41992  dihord2a  41993  dihord2b  41994  dihord11c  41998  dihlsscpre  42008  dihopelvalcpre  42022  dihlss  42024  dihord6apre  42030  dihord5b  42033  dihord5apre  42036