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Theorem dicvaddcl 41173
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 1135 . . 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 2735 . . . . . . 7 (Base‘𝑈) = (Base‘𝑈)
82, 3, 4, 5, 6, 7dicssdvh 41169 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (Base‘𝑈))
9 eqid 2735 . . . . . . . . 9 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10 eqid 2735 . . . . . . . . 9 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
114, 9, 10, 6, 7dvhvbase 41070 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
1211eqcomd 2741 . . . . . . 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, 13sseqtrrd 4037 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
15143adant3 1131 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝐼𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
16 simp3l 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (𝐼𝑄))
1715, 16sseldd 3996 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑋 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
18 simp3r 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (𝐼𝑄))
1915, 18sseldd 3996 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
20 eqid 2735 . . . 4 (Scalar‘𝑈) = (Scalar‘𝑈)
21 dicvaddcl.p . . . 4 + = (+g𝑈)
22 eqid 2735 . . . 4 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
234, 9, 10, 6, 20, 21, 22dvhvadd 41075 . . 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 837 . 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 41172 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
26253adant3r 1180 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑋) ∈ ((TEndo‘𝐾)‘𝑊))
272, 3, 4, 10, 5dicelval2nd 41172 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
28273adant3l 1179 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))
29 eqid 2735 . . . . . . . 8 (oc‘𝐾) = (oc‘𝐾)
302, 29, 3, 4lhpocnel 40001 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
31303ad2ant1 1132 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) 𝑊))
32 simp2 1136 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
33 eqid 2735 . . . . . . 7 (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
342, 3, 4, 9, 33ltrniotacl 40562 . . . . . 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 1370 . . . . 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 2735 . . . . . 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 41005 . . . . 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 1370 . . . 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 41067 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑠) ∘ (𝑡)))))
40393ad2ant1 1132 . . . . . 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 7448 . . . . 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 6909 . . . 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 2735 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
442, 3, 4, 43, 9, 5dicelval1sta 41170 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑋 ∈ (𝐼𝑄)) → (1st𝑋) = ((2nd𝑋)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45443adant3r 1180 . . . . 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 41170 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47463adant3l 1179 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (1st𝑌) = ((2nd𝑌)‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
4845, 47coeq12d 5878 . . . 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 2786 . . 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 40764 . . . . 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 1370 . . . 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 2839 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ ((TEndo‘𝐾)‘𝑊))
53 fvex 6920 . . . . . 6 (1st𝑋) ∈ V
54 fvex 6920 . . . . . 6 (1st𝑌) ∈ V
5553, 54coex 7953 . . . . 5 ((1st𝑋) ∘ (1st𝑌)) ∈ V
56 ovex 7464 . . . . 5 ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌)) ∈ V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 41160 . . . 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 1131 . . 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 713 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → ⟨((1st𝑋) ∘ (1st𝑌)), ((2nd𝑋)(+g‘(Scalar‘𝑈))(2nd𝑌))⟩ ∈ (𝐼𝑄))
6024, 59eqeltrd 2839 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  cmpt 5231   × cxp 5687  ccom 5693  cfv 6563  crio 7387  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301  lecple 17305  occoc 17306  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  DVecHcdvh 41061  DIsoCcdic 41155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-undef 8297  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tendo 40738  df-edring 40740  df-dvech 41062  df-dic 41156
This theorem is referenced by:  diclss  41176
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