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Theorem dicvaddcl 39759
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 1136 . . 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 39755 . . . . . 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 39656 . . . . . . . 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 481 . . . . . 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 4003 . . . . 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 1132 . . . 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 1201 . . . 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 3963 . . 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 1202 . . . 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 3963 . . 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 39661 . . 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 835 . 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 39758 . . . . . 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 1181 . . . . 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 39758 . . . . . 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 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‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
29 eqid 2731 . . . . . . . 8 (ocβ€˜πΎ) = (ocβ€˜πΎ)
302, 29, 3, 4lhpocnel 38587 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
31303ad2ant1 1133 . . . . . 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 1137 . . . . . 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 39148 . . . . . 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 1371 . . . . 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 39591 . . . . 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 1371 . . . 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 39653 . . . . . . 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 1133 . . . . . 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 7394 . . . . 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 6864 . . . 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 39756 . . . . . 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 1181 . . . . 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 39756 . . . . . 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 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‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
4845, 47coeq12d 5840 . . . 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 39350 . . . . 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 1371 . . . 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 6875 . . . . . 6 (1st β€˜π‘‹) ∈ V
54 fvex 6875 . . . . . 6 (1st β€˜π‘Œ) ∈ V
5553, 54coex 7887 . . . . 5 ((1st β€˜π‘‹) ∘ (1st β€˜π‘Œ)) ∈ V
56 ovex 7410 . . . . 5 ((2nd β€˜π‘‹)(+gβ€˜(Scalarβ€˜π‘ˆ))(2nd β€˜π‘Œ)) ∈ V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 39746 . . . 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 1132 . . 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 711 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3928  βŸ¨cop 4612   class class class wbr 5125   ↦ cmpt 5208   Γ— cxp 5651   ∘ ccom 5657  β€˜cfv 6516  β„©crio 7332  (class class class)co 7377   ∈ cmpo 7379  1st c1st 7939  2nd c2nd 7940  Basecbs 17109  +gcplusg 17162  Scalarcsca 17165  lecple 17169  occoc 17170  Atomscatm 37831  HLchlt 37918  LHypclh 38553  LTrncltrn 38670  TEndoctendo 39321  DVecHcdvh 39647  DIsoCcdic 39741
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-riotaBAD 37521
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4886  df-iun 4976  df-iin 4977  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-undef 8224  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-map 8789  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-3 12241  df-4 12242  df-5 12243  df-6 12244  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-struct 17045  df-slot 17080  df-ndx 17092  df-base 17110  df-plusg 17175  df-mulr 17176  df-sca 17178  df-vsca 17179  df-proset 18213  df-poset 18231  df-plt 18248  df-lub 18264  df-glb 18265  df-join 18266  df-meet 18267  df-p0 18343  df-p1 18344  df-lat 18350  df-clat 18417  df-oposet 37744  df-ol 37746  df-oml 37747  df-covers 37834  df-ats 37835  df-atl 37866  df-cvlat 37890  df-hlat 37919  df-llines 38067  df-lplanes 38068  df-lvols 38069  df-lines 38070  df-psubsp 38072  df-pmap 38073  df-padd 38365  df-lhyp 38557  df-laut 38558  df-ldil 38673  df-ltrn 38674  df-trl 38728  df-tendo 39324  df-edring 39326  df-dvech 39648  df-dic 39742
This theorem is referenced by:  diclss  39762
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