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Theorem dicvaddcl 36796
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 1081 . . 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 2651 . . . . . . 7 (Base‘𝑈) = (Base‘𝑈)
82, 3, 4, 5, 6, 7dicssdvh 36792 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (Base‘𝑈))
9 eqid 2651 . . . . . . . . 9 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10 eqid 2651 . . . . . . . . 9 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
114, 9, 10, 6, 7dvhvbase 36693 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1211eqcomd 2657 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
1312adantr 480 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
148, 13sseqtr4d 3675 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15143adant3 1101 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
16 simp3l 1109 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (𝐼𝑄))
1715, 16sseldd 3637 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18 simp3r 1110 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (𝐼𝑄))
1915, 18sseldd 3637 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20 eqid 2651 . . . 4 (Scalar‘𝑈) = (Scalar‘𝑈)
21 dicvaddcl.p . . . 4 + = (+g𝑈)
22 eqid 2651 . . . 4 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
234, 9, 10, 6, 20, 21, 22dvhvadd 36698 . . 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 1364 . 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 36795 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26253adant3r 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
272, 3, 4, 10, 5dicelval2nd 36795 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28273adant3l 1362 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29 eqid 2651 . . . . . . . 8 (oc‘𝐾) = (oc‘𝐾)
302, 29, 3, 4lhpocnel 35622 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
31303ad2ant1 1102 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
32 simp2 1082 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
33 eqid 2651 . . . . . . 7 (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
342, 3, 4, 9, 33ltrniotacl 36184 . . . . . 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 1366 . . . . 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 2651 . . . . . 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 36628 . . . . 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 1366 . . . 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 36690 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
40393ad2ant1 1102 . . . . . 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 6707 . . . . 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 6231 . . . 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 2651 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
442, 3, 4, 43, 9, 5dicelval1sta 36793 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (1st𝑋) = ((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45443adant3r 1363 . . . . 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 36793 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47463adant3l 1362 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
4845, 47coeq12d 5319 . . . 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 2696 . . 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 36386 . . . . 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 1366 . . . 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 2730 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
53 fvex 6239 . . . . . 6 (1st𝑋) ∈ V
54 fvex 6239 . . . . . 6 (1st𝑌) ∈ V
5553, 54coex 7160 . . . . 5 ((1st𝑋) ∘ (1st𝑌)) ∈ V
56 ovex 6718 . . . . 5 ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 36783 . . . 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 1101 . . 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 977 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩ ∈ (𝐼𝑄))
6024, 59eqeltrd 2730 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  ccom 5147  cfv 5926  crio 6650  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991  lecple 15995  occoc 15996  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  TEndoctendo 36357  DVecHcdvh 36684  DIsoCcdic 36778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360  df-edring 36362  df-dvech 36685  df-dic 36779
This theorem is referenced by:  diclss  36799
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