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Theorem dicvaddcl 40365
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 ≀ = (leβ€˜πΎ)
dicvaddcl.a 𝐴 = (Atomsβ€˜πΎ)
dicvaddcl.h 𝐻 = (LHypβ€˜πΎ)
dicvaddcl.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dicvaddcl.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
dicvaddcl.p + = (+gβ€˜π‘ˆ)
Assertion
Ref Expression
dicvaddcl (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (𝑋 + π‘Œ) ∈ (πΌβ€˜π‘„))

Proof of Theorem dicvaddcl
Dummy variables 𝑔 β„Ž 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1135 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 dicvaddcl.l . . . . . . 7 ≀ = (leβ€˜πΎ)
3 dicvaddcl.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
4 dicvaddcl.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
5 dicvaddcl.i . . . . . . 7 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
6 dicvaddcl.u . . . . . . 7 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
7 eqid 2731 . . . . . . 7 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
82, 3, 4, 5, 6, 7dicssdvh 40361 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) βŠ† (Baseβ€˜π‘ˆ))
9 eqid 2731 . . . . . . . . 9 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 eqid 2731 . . . . . . . . 9 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
114, 9, 10, 6, 7dvhvbase 40262 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
1211eqcomd 2737 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)) = (Baseβ€˜π‘ˆ))
1312adantr 480 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)) = (Baseβ€˜π‘ˆ))
148, 13sseqtrrd 4023 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) βŠ† (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
15143adant3 1131 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (πΌβ€˜π‘„) βŠ† (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
16 simp3l 1200 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ 𝑋 ∈ (πΌβ€˜π‘„))
1715, 16sseldd 3983 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ 𝑋 ∈ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
18 simp3r 1201 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ π‘Œ ∈ (πΌβ€˜π‘„))
1915, 18sseldd 3983 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ π‘Œ ∈ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))
20 eqid 2731 . . . 4 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
21 dicvaddcl.p . . . 4 + = (+gβ€˜π‘ˆ)
22 eqid 2731 . . . 4 (+gβ€˜(Scalarβ€˜π‘ˆ)) = (+gβ€˜(Scalarβ€˜π‘ˆ))
234, 9, 10, 6, 20, 21, 22dvhvadd 40267 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ π‘Œ ∈ (((LTrnβ€˜πΎ)β€˜π‘Š) Γ— ((TEndoβ€˜πΎ)β€˜π‘Š)))) β†’ (𝑋 + π‘Œ) = ⟨((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)), ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))⟩)
241, 17, 19, 23syl12anc 834 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (𝑋 + π‘Œ) = ⟨((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)), ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))⟩)
252, 3, 4, 10, 5dicelval2nd 40364 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑋 ∈ (πΌβ€˜π‘„)) β†’ (2nd β€˜π‘‹) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
26253adant3r 1180 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (2nd β€˜π‘‹) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
272, 3, 4, 10, 5dicelval2nd 40364 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
28273adant3l 1179 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
29 eqid 2731 . . . . . . . 8 (ocβ€˜πΎ) = (ocβ€˜πΎ)
302, 29, 3, 4lhpocnel 39193 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
31303ad2ant1 1132 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
32 simp2 1136 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
33 eqid 2731 . . . . . . 7 (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) = (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)
342, 3, 4, 9, 33ltrniotacl 39754 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
351, 31, 32, 34syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
36 eqid 2731 . . . . . 6 (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž)))) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))
379, 36tendospdi2 40197 . . . . 5 (((2nd β€˜π‘‹) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ∧ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ∧ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) = (((2nd β€˜π‘‹)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∘ ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄))))
3826, 28, 35, 37syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) = (((2nd β€˜π‘‹)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∘ ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄))))
394, 9, 10, 6, 20, 36, 22dvhfplusr 40259 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜(Scalarβ€˜π‘ˆ)) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž)))))
40393ad2ant1 1132 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (+gβ€˜(Scalarβ€˜π‘ˆ)) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž)))))
4140oveqd 7429 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) = ((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ)))
4241fveq1d 6893 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) = (((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
43 eqid 2731 . . . . . . 7 ((ocβ€˜πΎ)β€˜π‘Š) = ((ocβ€˜πΎ)β€˜π‘Š)
442, 3, 4, 43, 9, 5dicelval1sta 40362 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑋 ∈ (πΌβ€˜π‘„)) β†’ (1st β€˜π‘‹) = ((2nd β€˜π‘‹)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
45443adant3r 1180 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (1st β€˜π‘‹) = ((2nd β€˜π‘‹)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
462, 3, 4, 43, 9, 5dicelval1sta 40362 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
47463adant3l 1179 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
4845, 47coeq12d 5864 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) = (((2nd β€˜π‘‹)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∘ ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄))))
4938, 42, 483eqtr4rd 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) = (((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
504, 9, 10, 36tendoplcl 39956 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (2nd β€˜π‘‹) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ∧ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) β†’ ((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ)) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
511, 26, 28, 50syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ((2nd β€˜π‘‹)(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ((π‘ β€˜β„Ž) ∘ (π‘‘β€˜β„Ž))))(2nd β€˜π‘Œ)) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
5241, 51eqeltrd 2832 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
53 fvex 6904 . . . . . 6 (1st β€˜π‘‹) ∈ V
54 fvex 6904 . . . . . 6 (1st β€˜π‘Œ) ∈ V
5553, 54coex 7925 . . . . 5 ((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) ∈ V
56 ovex 7445 . . . . 5 ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) ∈ V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 40352 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (⟨((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)), ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))⟩ ∈ (πΌβ€˜π‘„) ↔ (((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) = (((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
58573adant3 1131 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (⟨((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)), ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))⟩ ∈ (πΌβ€˜π‘„) ↔ (((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) = (((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
5949, 52, 58mpbir2and 710 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ ⟨((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)), ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ))⟩ ∈ (πΌβ€˜π‘„))
6024, 59eqeltrd 2832 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑋 ∈ (πΌβ€˜π‘„) ∧ π‘Œ ∈ (πΌβ€˜π‘„))) β†’ (𝑋 + π‘Œ) ∈ (πΌβ€˜π‘„))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  βŸ¨cop 4634   class class class wbr 5148   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680  β€˜cfv 6543  β„©crio 7367  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7977  2nd c2nd 7978  Basecbs 17149  +gcplusg 17202  Scalarcsca 17205  lecple 17209  occoc 17210  Atomscatm 38437  HLchlt 38524  LHypclh 39159  LTrncltrn 39276  TEndoctendo 39927  DVecHcdvh 40253  DIsoCcdic 40347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-riotaBAD 38127
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-undef 8262  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-struct 17085  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334  df-tendo 39930  df-edring 39932  df-dvech 40254  df-dic 40348
This theorem is referenced by:  diclss  40368
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