Theorem diblss 41137

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 2730	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diblss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2729	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diblss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2729	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2729	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 41050	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2735	. . 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 2729	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2729	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 41054	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2735	. . 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 2730	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (+g‘𝑈) = (+g‘𝑈))
16		eqidd 2730	. 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 41136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3981	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dibn0 41120	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅)
25		fvex 6853	. . . . . . 7 ⊢ (𝑥‘(1st ‘𝑎)) ∈ V
26		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
27		fvex 6853	. . . . . . . 8 ⊢ (2nd ‘𝑎) ∈ V
28	26, 27	coex 7886	. . . . . . 7 ⊢ (𝑥 ∘ (2nd ‘𝑎)) ∈ V
29	25, 28	op1st 7955	. . . . . 6 ⊢ (1st ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥‘(1st ‘𝑎))
30	29	coeq1i 5813	. . . . 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 41117	. . . . . . . . 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 40734	. . . . . . . 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 41117	. . . . . . . 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 40686	. . . . . . 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 39330	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑋) ∧ 𝑏 ∈ (𝐼‘𝑋))) → 𝐾 ∈ Lat)
46		eqid 2729	. . . . . . . . 9 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
47	19, 2, 10, 46	trlcl 40131	. . . . . . . 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 40131	. . . . . . . . 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 40131	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
54	19, 53	latjcl 18374	. . . . . . . 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 40694	. . . . . . . 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 40131	. . . . . . . . . 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 40728	. . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
64	19, 20, 2, 63, 21	dibelval1st 41116	. . . . . . . . . . 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 41011	. . . . . . . . . 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 18381	. . . . . . . 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 41116	. . . . . . . . . 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 41011	. . . . . . . . 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 18385	. . . . . . . . 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 18381	. . . . . 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 41000	. . . . . . 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 2832	. . . 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 2729	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
83	2, 10, 3, 4, 5, 81, 82	dvhfplusr 41051	. . . . . . . 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 7956	. . . . . . . 8 ⊢ (2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩) = (𝑥 ∘ (2nd ‘𝑎))
86		eqid 2729	. . . . . . . . . . . 12 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
87	19, 20, 2, 10, 86, 21	dibelval2nd 41119	. . . . . . . . . . 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 5816	. . . . . . . . 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 40794	. . . . . . . . . 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 2764	. . . . . . . 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 2776	. . . . . . 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 41119	. . . . . . . 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 7390	. . . . . 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 40757	. . . . . . . 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 40758	. . . . . . 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 2764	. . . . 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 7402	. . . . . 6 ⊢ ((2nd ‘⟨(𝑥‘(1st ‘𝑎)), (𝑥 ∘ (2nd ‘𝑎))⟩)(+g‘(Scalar‘𝑈))(2nd ‘𝑏)) ∈ V
104	103	elsn 4600	. . . . 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 5668	. . . 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 3944	. . . . . 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 2729	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
111	2, 10, 3, 4, 110	dvhvsca 41068	. . . . . 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 7384	. . . 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 2828	. . . . . . 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 40739	. . . . . . 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 5668	. . . . . 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 3944	. . . . 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 2729	. . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈)
121	2, 10, 3, 4, 5, 120, 82	dvhvadd 41059	. . . . 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 2764	. . 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 41111	. . . 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 2843	. 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 20817	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 2109   ⊆ wss 3911  {csn 4585  ⟨cop 4591   class class class wbr 5102   ↦ cmpt 5183   I cid 5525   × cxp 5629   ↾ cres 5633   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  +_gcplusg 17196  Scalarcsca 17199   ·_𝑠 cvsca 17200  lecple 17203  joincjn 18248  Latclat 18366  LSubSpclss 20813  HLchlt 39316  LHypclh 39951  LTrncltrn 40068  trLctrl 40125  TEndoctendo 40719  DIsoAcdia 40995  DVecHcdvh 41045  DIsoBcdib 41105

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-riotaBAD 38919

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-undef 8229  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18367  df-clat 18434  df-lss 20814  df-oposet 39142  df-ol 39144  df-oml 39145  df-covers 39232  df-ats 39233  df-atl 39264  df-cvlat 39288  df-hlat 39317  df-llines 39465  df-lplanes 39466  df-lvols 39467  df-lines 39468  df-psubsp 39470  df-pmap 39471  df-padd 39763  df-lhyp 39955  df-laut 39956  df-ldil 40071  df-ltrn 40072  df-trl 40126  df-tendo 40722  df-edring 40724  df-disoa 40996  df-dvech 41046  df-dib 41106

This theorem is referenced by:  diblsmopel  41138  cdlemn5pre  41167  cdlemn11c  41176  dihjustlem  41183  dihord1  41185  dihord2a  41186  dihord2b  41187  dihord11c  41191  dihlsscpre  41201  dihopelvalcpre  41215  dihlss  41217  dihord6apre  41223  dihord5b  41226  dihord5apre  41229