dicvaddcl - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dicvaddcl		Structured version Visualization version GIF version

Theorem dicvaddcl 40364

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 1134	. . 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 2730	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
8	2, 3, 4, 5, 6, 7	dicssdvh 40360	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
9		eqid 2730	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2730	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
11	4, 9, 10, 6, 7	dvhvbase 40261	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
12	11	eqcomd 2736	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
13	12	adantr 479	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	8, 13	sseqtrrd 4022	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15	14	3adant3 1130	. . . 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 1199	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (𝐼‘𝑄))
17	15, 16	sseldd 3982	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18		simp3r 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (𝐼‘𝑄))
19	15, 18	sseldd 3982	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20		eqid 2730	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
21		dicvaddcl.p	. . . 4 ⊢ + = (+_g‘𝑈)
22		eqid 2730	. . . 4 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
23	4, 9, 10, 6, 20, 21, 22	dvhvadd 40266	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ∧ 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑋 + 𝑌) = ⟨((1^st ‘𝑋) ∘ (1^st ‘𝑌)), ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))⟩)
24	1, 17, 19, 23	syl12anc 833	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) = ⟨((1^st ‘𝑋) ∘ (1^st ‘𝑌)), ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))⟩)
25	2, 3, 4, 10, 5	dicelval2nd 40363	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑋 ∈ (𝐼‘𝑄)) → (2^nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26	25	3adant3r 1179	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2^nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
27	2, 3, 4, 10, 5	dicelval2nd 40363	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (2^nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28	27	3adant3l 1178	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2^nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29		eqid 2730	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
30	2, 29, 3, 4	lhpocnel 39192	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
31	30	3ad2ant1 1131	. . . . . 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 1135	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
33		eqid 2730	. . . . . . 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 39753	. . . . . 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 1369	. . . . 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 2730	. . . . . 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 40196	. . . . 5 ⊢ (((2^nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2^nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊)) → (((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2^nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2^nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
38	26, 28, 35, 37	syl3anc 1369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2^nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2^nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
39	4, 9, 10, 6, 20, 36, 22	dvhfplusr 40258	. . . . . . 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 1131	. . . . . 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 7428	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌)) = ((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌)))
42	41	fveq1d 6892	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
43		eqid 2730	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
44	2, 3, 4, 43, 9, 5	dicelval1sta 40361	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑋 ∈ (𝐼‘𝑄)) → (1^st ‘𝑋) = ((2^nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45	44	3adant3r 1179	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (1^st ‘𝑋) = ((2^nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
46	2, 3, 4, 43, 9, 5	dicelval1sta 40361	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47	46	3adant3l 1178	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
48	45, 47	coeq12d 5863	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((1^st ‘𝑋) ∘ (1^st ‘𝑌)) = (((2^nd ‘𝑋)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2^nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
49	38, 42, 48	3eqtr4rd 2781	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((1^st ‘𝑋) ∘ (1^st ‘𝑌)) = (((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
50	4, 9, 10, 36	tendoplcl 39955	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2^nd ‘𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2^nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)) → ((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
51	1, 26, 28, 50	syl3anc 1369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2^nd ‘𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠‘ℎ) ∘ (𝑡‘ℎ))))(2^nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
52	41, 51	eqeltrd 2831	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
53		fvex 6903	. . . . . 6 ⊢ (1^st ‘𝑋) ∈ V
54		fvex 6903	. . . . . 6 ⊢ (1^st ‘𝑌) ∈ V
55	53, 54	coex 7923	. . . . 5 ⊢ ((1^st ‘𝑋) ∘ (1^st ‘𝑌)) ∈ V
56		ovex 7444	. . . . 5 ⊢ ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌)) ∈ V
57	2, 3, 4, 43, 9, 10, 5, 55, 56	dicopelval 40351	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨((1^st ‘𝑋) ∘ (1^st ‘𝑌)), ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ (((1^st ‘𝑋) ∘ (1^st ‘𝑌)) = (((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
58	57	3adant3 1130	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (⟨((1^st ‘𝑋) ∘ (1^st ‘𝑌)), ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ (((1^st ‘𝑋) ∘ (1^st ‘𝑌)) = (((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
59	49, 52, 58	mpbir2and 709	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → ⟨((1^st ‘𝑋) ∘ (1^st ‘𝑌)), ((2^nd ‘𝑋)(+_g‘(Scalar‘𝑈))(2^nd ‘𝑌))⟩ ∈ (𝐼‘𝑄))
60	24, 59	eqeltrd 2831	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ (𝐼‘𝑄) ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 + 𝑌) ∈ (𝐼‘𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 ⟨cop 4633 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ∘ ccom 5679 ‘cfv 6542 ℩crio 7366 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7975 2^nd c2nd 7976 Basecbs 17148 +_gcplusg 17201 Scalarcsca 17204 lecple 17208 occoc 17209 Atomscatm 38436 HLchlt 38523 LHypclh 39158 LTrncltrn 39275 TEndoctendo 39926 DVecHcdvh 40252 DIsoCcdic 40346

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38126

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tendo 39929 df-edring 39931 df-dvech 40253 df-dic 40347

This theorem is referenced by: diclss 40367

W3C validator