Theorem dicvaddcl 41184

Description: Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Hypotheses

Ref	Expression
dicvaddcl.l	⊢ ≤ = (le‘𝐾)
dicvaddcl.a	⊢ 𝐴 = (Atoms‘𝐾)
dicvaddcl.h	⊢ 𝐻 = (LHyp‘𝐾)
dicvaddcl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dicvaddcl.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicvaddcl.p	⊢ + = (+g‘𝑈)

Assertion

Ref	Expression
dicvaddcl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) ∈ (𝐼‘𝑄))

Proof of Theorem dicvaddcl

Dummy variables 𝑔 ℎ 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dicvaddcl.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
3		dicvaddcl.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		dicvaddcl.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
5		dicvaddcl.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
6		dicvaddcl.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
8	2, 3, 4, 5, 6, 7	dicssdvh 41180	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
9		eqid 2729	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2729	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
11	4, 9, 10, 6, 7	dvhvbase 41081	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
12	11	eqcomd 2735	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
13	12	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	8, 13	sseqtrrd 3984	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15	14	3adant3 1132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
16		simp3l 1202	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (𝐼‘𝑄))
17	15, 16	sseldd 3947	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18		simp3r 1203	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (𝐼‘𝑄))
19	15, 18	sseldd 3947	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20		eqid 2729	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
21		dicvaddcl.p	. . . 4 ⊢ + = (+g‘𝑈)
22		eqid 2729	. . . 4 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
23	4, 9, 10, 6, 20, 21, 22	dvhvadd 41086	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ∧ 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑋 + 𝑌) = ⟨((1st ‘𝑋) ∘ (1st ‘𝑌)), ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))⟩)
24	1, 17, 19, 23	syl12anc 836	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) = ⟨((1st ‘𝑋) ∘ (1st ‘𝑌)), ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))⟩)
25	2, 3, 4, 10, 5	dicelval2nd 41183	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑋 ∈ (𝐼‘𝑄)) → (2nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26	25	3adant3r 1182	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
27	2, 3, 4, 10, 5	dicelval2nd 41183	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28	27	3adant3l 1181	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29		eqid 2729	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
30	2, 29, 3, 4	lhpocnel 40012	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
31	30	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
32		simp2 1137	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
33		eqid 2729	. . . . . . 7 ⊢ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
34	2, 3, 4, 9, 33	ltrniotacl 40573	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
35	1, 31, 32, 34	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
36		eqid 2729	. . . . . 6 ⊢ (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))))
37	9, 36	tendospdi2 41016	. . . . 5 ⊢ (((2nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊)) → (((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
38	26, 28, 35, 37	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
39	4, 9, 10, 6, 20, 36, 22	dvhfplusr 41078	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ)))))
40	39	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ)))))
41	40	oveqd 7404	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) = ((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌)))
42	41	fveq1d 6860	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
43		eqid 2729	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
44	2, 3, 4, 43, 9, 5	dicelval1sta 41181	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑋 ∈ (𝐼‘𝑄)) → (1st ‘𝑋) = ((2nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45	44	3adant3r 1182	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (1st ‘𝑋) = ((2nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
46	2, 3, 4, 43, 9, 5	dicelval1sta 41181	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47	46	3adant3l 1181	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
48	45, 47	coeq12d 5828	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((1st ‘𝑋) ∘ (1st ‘𝑌)) = (((2nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
49	38, 42, 48	3eqtr4rd 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((1st ‘𝑋) ∘ (1st ‘𝑌)) = (((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
50	4, 9, 10, 36	tendoplcl 40775	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)) → ((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
51	1, 26, 28, 50	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
52	41, 51	eqeltrd 2828	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
53		fvex 6871	. . . . . 6 ⊢ (1st ‘𝑋) ∈ V
54		fvex 6871	. . . . . 6 ⊢ (1st ‘𝑌) ∈ V
55	53, 54	coex 7906	. . . . 5 ⊢ ((1st ‘𝑋) ∘ (1st ‘𝑌)) ∈ V
56		ovex 7420	. . . . 5 ⊢ ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) ∈ V
57	2, 3, 4, 43, 9, 10, 5, 55, 56	dicopelval 41171	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨((1st ‘𝑋) ∘ (1st ‘𝑌)), ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ (((1st ‘𝑋) ∘ (1st ‘𝑌)) = (((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
58	57	3adant3 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (⟨((1st ‘𝑋) ∘ (1st ‘𝑌)), ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ (((1st ‘𝑋) ∘ (1st ‘𝑌)) = (((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
59	49, 52, 58	mpbir2and 713	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ⟨((1st ‘𝑋) ∘ (1st ‘𝑌)), ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄))
60	24, 59	eqeltrd 2828	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) ∈ (𝐼‘𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ⟨cop 4595   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636   ∘ ccom 5642  ‘cfv 6511  ℩crio 7343  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  +_gcplusg 17220  Scalarcsca 17223  lecple 17227  occoc 17228  Atomscatm 39256  HLchlt 39343  LHypclh 39978  LTrncltrn 40095  TEndoctendo 40746  DVecHcdvh 41072  DIsoCcdic 41166

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-riotaBAD 38946

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-undef 8252  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494  df-lines 39495  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099  df-trl 40153  df-tendo 40749  df-edring 40751  df-dvech 41073  df-dic 41167

This theorem is referenced by:  diclss  41187