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

Theorem dicvaddcl 38444
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 + = (+g𝑈)
Assertion
Ref Expression
dicvaddcl (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑋 + 𝑌) ∈ (𝐼𝑄))

Proof of Theorem dicvaddcl
Dummy variables 𝑔 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
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 2822 . . . . . . 7 (Base‘𝑈) = (Base‘𝑈)
82, 3, 4, 5, 6, 7dicssdvh 38440 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (Base‘𝑈))
9 eqid 2822 . . . . . . . . 9 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10 eqid 2822 . . . . . . . . 9 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
114, 9, 10, 6, 7dvhvbase 38341 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1211eqcomd 2828 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
1312adantr 484 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
148, 13sseqtrrd 3983 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15143adant3 1129 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
16 simp3l 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (𝐼𝑄))
1715, 16sseldd 3943 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18 simp3r 1199 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (𝐼𝑄))
1915, 18sseldd 3943 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20 eqid 2822 . . . 4 (Scalar‘𝑈) = (Scalar‘𝑈)
21 dicvaddcl.p . . . 4 + = (+g𝑈)
22 eqid 2822 . . . 4 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
234, 9, 10, 6, 20, 21, 22dvhvadd 38346 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ∧ 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))) → (𝑋 + 𝑌) = ⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩)
241, 17, 19, 23syl12anc 835 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑋 + 𝑌) = ⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩)
252, 3, 4, 10, 5dicelval2nd 38443 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26253adant3r 1178 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
272, 3, 4, 10, 5dicelval2nd 38443 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28273adant3l 1177 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29 eqid 2822 . . . . . . . 8 (oc‘𝐾) = (oc‘𝐾)
302, 29, 3, 4lhpocnel 37272 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
31303ad2ant1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
32 simp2 1134 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
33 eqid 2822 . . . . . . 7 (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
342, 3, 4, 9, 33ltrniotacl 37833 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
351, 31, 32, 34syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
36 eqid 2822 . . . . . 6 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))
379, 36tendospdi2 38276 . . . . 5 (((2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊)) → (((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
3826, 28, 35, 37syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
394, 9, 10, 6, 20, 36, 22dvhfplusr 38338 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
40393ad2ant1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
4140oveqd 7157 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) = ((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌)))
4241fveq1d 6654 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
43 eqid 2822 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
442, 3, 4, 43, 9, 5dicelval1sta 38441 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (1st𝑋) = ((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45443adant3r 1178 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (1st𝑋) = ((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
462, 3, 4, 43, 9, 5dicelval1sta 38441 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47463adant3l 1177 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
4845, 47coeq12d 5712 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((1st𝑋) ∘ (1st𝑌)) = (((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∘ ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
4938, 42, 483eqtr4rd 2868 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((1st𝑋) ∘ (1st𝑌)) = (((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
504, 9, 10, 36tendoplcl 38035 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)) → ((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
511, 26, 28, 50syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((2nd𝑋)(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡))))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
5241, 51eqeltrd 2914 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
53 fvex 6665 . . . . . 6 (1st𝑋) ∈ V
54 fvex 6665 . . . . . 6 (1st𝑌) ∈ V
5553, 54coex 7621 . . . . 5 ((1st𝑋) ∘ (1st𝑌)) ∈ V
56 ovex 7173 . . . . 5 ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 38431 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩ ∈ (𝐼𝑄) ↔ (((1st𝑋) ∘ (1st𝑌)) = (((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
58573adant3 1129 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩ ∈ (𝐼𝑄) ↔ (((1st𝑋) ∘ (1st𝑌)) = (((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))))
5949, 52, 58mpbir2and 712 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩ ∈ (𝐼𝑄))
6024, 59eqeltrd 2914 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑋 + 𝑌) ∈ (𝐼𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wss 3908  cop 4545   class class class wbr 5042  cmpt 5122   × cxp 5530  ccom 5536  cfv 6334  crio 7097  (class class class)co 7140  cmpo 7142  1st c1st 7673  2nd c2nd 7674  Basecbs 16474  +gcplusg 16556  Scalarcsca 16559  lecple 16563  occoc 16564  Atomscatm 36517  HLchlt 36604  LHypclh 37238  LTrncltrn 37355  TEndoctendo 38006  DVecHcdvh 38332  DIsoCcdic 38426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-riotaBAD 36207
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-undef 7926  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-llines 36752  df-lplanes 36753  df-lvols 36754  df-lines 36755  df-psubsp 36757  df-pmap 36758  df-padd 37050  df-lhyp 37242  df-laut 37243  df-ldil 37358  df-ltrn 37359  df-trl 37413  df-tendo 38009  df-edring 38011  df-dvech 38333  df-dic 38427
This theorem is referenced by:  diclss  38447
  Copyright terms: Public domain W3C validator