Theorem dicvaddcl 42024

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 1154	. . 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 2765	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
8	2, 3, 4, 5, 6, 7	dicssdvh 42020	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
9		eqid 2765	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2765	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
11	4, 9, 10, 6, 7	dvhvbase 41921	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
12	11	eqcomd 2771	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
13	12	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	8, 13	sseqtrrd 3975	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15	14	3adant3 1150	. . . 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 1220	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (𝐼‘𝑄))
17	15, 16	sseldd 3939	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18		simp3r 1221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (𝐼‘𝑄))
19	15, 18	sseldd 3939	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20		eqid 2765	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
21		dicvaddcl.p	. . . 4 ⊢ + = (+g‘𝑈)
22		eqid 2765	. . . 4 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
23	4, 9, 10, 6, 20, 21, 22	dvhvadd 41926	. . 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 850	. 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 42023	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑋 ∈ (𝐼‘𝑄)) → (2nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26	25	3adant3r 1200	. . . . 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 42023	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28	27	3adant3l 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29		eqid 2765	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
30	2, 29, 3, 4	lhpocnel 40852	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
31	30	3ad2ant1 1151	. . . . . 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 1155	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
33		eqid 2765	. . . . . . 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 41413	. . . . . 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 1398	. . . . 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 2765	. . . . . 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 41856	. . . . 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 1398	. . . 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 41918	. . . . . . 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 1151	. . . . . 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 7436	. . . . 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 6887	. . . 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 2765	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
44	2, 3, 4, 43, 9, 5	dicelval1sta 42021	. . . . . 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 1200	. . . . 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 42021	. . . . . 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 1199	. . . . 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 5852	. . . 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 2811	. . 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 41615	. . . . 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 1398	. . . 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 2865	. . 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 6898	. . . . . 6 ⊢ (1st ‘𝑋) ∈ V
54		fvex 6898	. . . . . 6 ⊢ (1st ‘𝑌) ∈ V
55	53, 54	coex 7933	. . . . 5 ⊢ ((1st ‘𝑋) ∘ (1st ‘𝑌)) ∈ V
56		ovex 7452	. . . . 5 ⊢ ((2nd ‘𝑋)(+g‘(Scalar‘𝑈))(2nd ‘𝑌)) ∈ V
57	2, 3, 4, 43, 9, 10, 5, 55, 56	dicopelval 42011	. . . 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 1150	. . 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 726	. 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 2865	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) ∈ (𝐼‘𝑄))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   ∘ ccom 5667  ‘cfv 6540  ℩crio 7375  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17293  +_gcplusg 17334  Scalarcsca 17337  lecple 17341  occoc 17342  Atomscatm 40097  HLchlt 40184  LHypclh 40818  LTrncltrn 40935  TEndoctendo 41586  DVecHcdvh 41912  DIsoCcdic 42006

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-riotaBAD 39787

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-undef 8275  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-n0 12522  df-z 12609  df-uz 12881  df-fz 13554  df-struct 17231  df-slot 17266  df-ndx 17278  df-base 17294  df-plusg 17347  df-mulr 17348  df-sca 17350  df-vsca 17351  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-p1 18504  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-llines 40332  df-lplanes 40333  df-lvols 40334  df-lines 40335  df-psubsp 40337  df-pmap 40338  df-padd 40630  df-lhyp 40822  df-laut 40823  df-ldil 40938  df-ltrn 40939  df-trl 40993  df-tendo 41589  df-edring 41591  df-dvech 41913  df-dic 42007

This theorem is used by:  diclss  42027