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Theorem dvhvaddass 39563
Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Hypotheses
Ref Expression
dvhvaddcl.h 𝐻 = (LHypβ€˜πΎ)
dvhvaddcl.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhvaddcl.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhvaddcl.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dvhvaddcl.d 𝐷 = (Scalarβ€˜π‘ˆ)
dvhvaddcl.p ⨣ = (+gβ€˜π·)
dvhvaddcl.a + = (+gβ€˜π‘ˆ)
Assertion
Ref Expression
dvhvaddass (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))

Proof of Theorem dvhvaddass
StepHypRef Expression
1 coass 6218 . . . 4 (((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∘ (1st β€˜πΌ)) = ((1st β€˜πΉ) ∘ ((1st β€˜πΊ) ∘ (1st β€˜πΌ)))
2 dvhvaddcl.h . . . . . . . . 9 𝐻 = (LHypβ€˜πΎ)
3 dvhvaddcl.t . . . . . . . . 9 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
4 dvhvaddcl.e . . . . . . . . 9 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
5 dvhvaddcl.u . . . . . . . . 9 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
6 dvhvaddcl.d . . . . . . . . 9 𝐷 = (Scalarβ€˜π‘ˆ)
7 dvhvaddcl.a . . . . . . . . 9 + = (+gβ€˜π‘ˆ)
8 dvhvaddcl.p . . . . . . . . 9 ⨣ = (+gβ€˜π·)
92, 3, 4, 5, 6, 7, 8dvhvadd 39558 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + 𝐺) = ⟨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩)
1093adantr3 1172 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + 𝐺) = ⟨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩)
1110fveq2d 6847 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝐹 + 𝐺)) = (1st β€˜βŸ¨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩))
12 fvex 6856 . . . . . . . 8 (1st β€˜πΉ) ∈ V
13 fvex 6856 . . . . . . . 8 (1st β€˜πΊ) ∈ V
1412, 13coex 7868 . . . . . . 7 ((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∈ V
15 ovex 7391 . . . . . . 7 ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ∈ V
1614, 15op1st 7930 . . . . . 6 (1st β€˜βŸ¨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩) = ((1st β€˜πΉ) ∘ (1st β€˜πΊ))
1711, 16eqtrdi 2793 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝐹 + 𝐺)) = ((1st β€˜πΉ) ∘ (1st β€˜πΊ)))
1817coeq1d 5818 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((1st β€˜(𝐹 + 𝐺)) ∘ (1st β€˜πΌ)) = (((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∘ (1st β€˜πΌ)))
192, 3, 4, 5, 6, 7, 8dvhvadd 39558 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐺 + 𝐼) = ⟨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩)
20193adantr1 1170 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐺 + 𝐼) = ⟨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩)
2120fveq2d 6847 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝐺 + 𝐼)) = (1st β€˜βŸ¨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩))
22 fvex 6856 . . . . . . . 8 (1st β€˜πΌ) ∈ V
2313, 22coex 7868 . . . . . . 7 ((1st β€˜πΊ) ∘ (1st β€˜πΌ)) ∈ V
24 ovex 7391 . . . . . . 7 ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ)) ∈ V
2523, 24op1st 7930 . . . . . 6 (1st β€˜βŸ¨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩) = ((1st β€˜πΊ) ∘ (1st β€˜πΌ))
2621, 25eqtrdi 2793 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝐺 + 𝐼)) = ((1st β€˜πΊ) ∘ (1st β€˜πΌ)))
2726coeq2d 5819 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((1st β€˜πΉ) ∘ (1st β€˜(𝐺 + 𝐼))) = ((1st β€˜πΉ) ∘ ((1st β€˜πΊ) ∘ (1st β€˜πΌ))))
281, 18, 273eqtr4a 2803 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((1st β€˜(𝐹 + 𝐺)) ∘ (1st β€˜πΌ)) = ((1st β€˜πΉ) ∘ (1st β€˜(𝐺 + 𝐼))))
29 xp2nd 7955 . . . . . 6 (𝐹 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΉ) ∈ 𝐸)
30 xp2nd 7955 . . . . . 6 (𝐺 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΊ) ∈ 𝐸)
31 xp2nd 7955 . . . . . 6 (𝐼 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΌ) ∈ 𝐸)
3229, 30, 313anim123i 1152 . . . . 5 ((𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸)) β†’ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸))
33 eqid 2737 . . . . . . . . . 10 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
342, 33, 5, 6dvhsca 39548 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
352, 33erngdv 39459 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
3634, 35eqeltrd 2838 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
37 drnggrp 20196 . . . . . . . 8 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
3938adantr 482 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ 𝐷 ∈ Grp)
40 simpr1 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΉ) ∈ 𝐸)
41 eqid 2737 . . . . . . . . 9 (Baseβ€˜π·) = (Baseβ€˜π·)
422, 4, 5, 6, 41dvhbase 39549 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
4342adantr 482 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (Baseβ€˜π·) = 𝐸)
4440, 43eleqtrrd 2841 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΉ) ∈ (Baseβ€˜π·))
45 simpr2 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΊ) ∈ 𝐸)
4645, 43eleqtrrd 2841 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΊ) ∈ (Baseβ€˜π·))
47 simpr3 1197 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΌ) ∈ 𝐸)
4847, 43eleqtrrd 2841 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (2nd β€˜πΌ) ∈ (Baseβ€˜π·))
4941, 8grpass 18758 . . . . . 6 ((𝐷 ∈ Grp ∧ ((2nd β€˜πΉ) ∈ (Baseβ€˜π·) ∧ (2nd β€˜πΊ) ∈ (Baseβ€˜π·) ∧ (2nd β€˜πΌ) ∈ (Baseβ€˜π·))) β†’ (((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ⨣ (2nd β€˜πΌ)) = ((2nd β€˜πΉ) ⨣ ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))))
5039, 44, 46, 48, 49syl13anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜πΉ) ∈ 𝐸 ∧ (2nd β€˜πΊ) ∈ 𝐸 ∧ (2nd β€˜πΌ) ∈ 𝐸)) β†’ (((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ⨣ (2nd β€˜πΌ)) = ((2nd β€˜πΉ) ⨣ ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))))
5132, 50sylan2 594 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ⨣ (2nd β€˜πΌ)) = ((2nd β€˜πΉ) ⨣ ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))))
5210fveq2d 6847 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝐹 + 𝐺)) = (2nd β€˜βŸ¨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩))
5314, 15op2nd 7931 . . . . . 6 (2nd β€˜βŸ¨((1st β€˜πΉ) ∘ (1st β€˜πΊ)), ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))⟩) = ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ))
5452, 53eqtrdi 2793 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝐹 + 𝐺)) = ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)))
5554oveq1d 7373 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜(𝐹 + 𝐺)) ⨣ (2nd β€˜πΌ)) = (((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ⨣ (2nd β€˜πΌ)))
5620fveq2d 6847 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝐺 + 𝐼)) = (2nd β€˜βŸ¨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩))
5723, 24op2nd 7931 . . . . . 6 (2nd β€˜βŸ¨((1st β€˜πΊ) ∘ (1st β€˜πΌ)), ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))⟩) = ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))
5856, 57eqtrdi 2793 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝐺 + 𝐼)) = ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ)))
5958oveq2d 7374 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜πΉ) ⨣ (2nd β€˜(𝐺 + 𝐼))) = ((2nd β€˜πΉ) ⨣ ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ))))
6051, 55, 593eqtr4d 2787 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜(𝐹 + 𝐺)) ⨣ (2nd β€˜πΌ)) = ((2nd β€˜πΉ) ⨣ (2nd β€˜(𝐺 + 𝐼))))
6128, 60opeq12d 4839 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ⟨((1st β€˜(𝐹 + 𝐺)) ∘ (1st β€˜πΌ)), ((2nd β€˜(𝐹 + 𝐺)) ⨣ (2nd β€˜πΌ))⟩ = ⟨((1st β€˜πΉ) ∘ (1st β€˜(𝐺 + 𝐼))), ((2nd β€˜πΉ) ⨣ (2nd β€˜(𝐺 + 𝐼)))⟩)
62 simpl 484 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
632, 3, 4, 5, 6, 8, 7dvhvaddcl 39561 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + 𝐺) ∈ (𝑇 Γ— 𝐸))
64633adantr3 1172 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + 𝐺) ∈ (𝑇 Γ— 𝐸))
65 simpr3 1197 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐼 ∈ (𝑇 Γ— 𝐸))
662, 3, 4, 5, 6, 7, 8dvhvadd 39558 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝐹 + 𝐺) ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝐹 + 𝐺) + 𝐼) = ⟨((1st β€˜(𝐹 + 𝐺)) ∘ (1st β€˜πΌ)), ((2nd β€˜(𝐹 + 𝐺)) ⨣ (2nd β€˜πΌ))⟩)
6762, 64, 65, 66syl12anc 836 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝐹 + 𝐺) + 𝐼) = ⟨((1st β€˜(𝐹 + 𝐺)) ∘ (1st β€˜πΌ)), ((2nd β€˜(𝐹 + 𝐺)) ⨣ (2nd β€˜πΌ))⟩)
68 simpr1 1195 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐹 ∈ (𝑇 Γ— 𝐸))
692, 3, 4, 5, 6, 8, 7dvhvaddcl 39561 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐺 + 𝐼) ∈ (𝑇 Γ— 𝐸))
70693adantr1 1170 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐺 + 𝐼) ∈ (𝑇 Γ— 𝐸))
712, 3, 4, 5, 6, 7, 8dvhvadd 39558 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ (𝐺 + 𝐼) ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + (𝐺 + 𝐼)) = ⟨((1st β€˜πΉ) ∘ (1st β€˜(𝐺 + 𝐼))), ((2nd β€˜πΉ) ⨣ (2nd β€˜(𝐺 + 𝐼)))⟩)
7262, 68, 70, 71syl12anc 836 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + (𝐺 + 𝐼)) = ⟨((1st β€˜πΉ) ∘ (1st β€˜(𝐺 + 𝐼))), ((2nd β€˜πΉ) ⨣ (2nd β€˜(𝐺 + 𝐼)))⟩)
7361, 67, 723eqtr4d 2787 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   Γ— cxp 5632   ∘ ccom 5638  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Basecbs 17084  +gcplusg 17134  Scalarcsca 17137  Grpcgrp 18749  DivRingcdr 20186  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  TEndoctendo 39218  EDRingcedring 39219  DVecHcdvh 39544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-riotaBAD 37418
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-undef 8205  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-n0 12415  df-z 12501  df-uz 12765  df-fz 13426  df-struct 17020  df-sets 17037  df-slot 17055  df-ndx 17067  df-base 17085  df-ress 17114  df-plusg 17147  df-mulr 17148  df-sca 17150  df-vsca 17151  df-0g 17324  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-mgp 19898  df-ur 19915  df-ring 19967  df-oppr 20050  df-dvdsr 20071  df-unit 20072  df-invr 20102  df-dvr 20113  df-drng 20188  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-llines 37964  df-lplanes 37965  df-lvols 37966  df-lines 37967  df-psubsp 37969  df-pmap 37970  df-padd 38262  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625  df-tendo 39221  df-edring 39223  df-dvech 39545
This theorem is referenced by:  dvhgrp  39573
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