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Theorem dvhvaddass 39957
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 6262 . . . 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 39952 . . . . . . . 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 6893 . . . . . 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 6902 . . . . . . . 8 (1st β€˜πΉ) ∈ V
13 fvex 6902 . . . . . . . 8 (1st β€˜πΊ) ∈ V
1412, 13coex 7918 . . . . . . 7 ((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∈ V
15 ovex 7439 . . . . . . 7 ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ∈ V
1614, 15op1st 7980 . . . . . 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 2789 . . . . 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 5860 . . . 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 39952 . . . . . . . 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 6893 . . . . . 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 6902 . . . . . . . 8 (1st β€˜πΌ) ∈ V
2313, 22coex 7918 . . . . . . 7 ((1st β€˜πΊ) ∘ (1st β€˜πΌ)) ∈ V
24 ovex 7439 . . . . . . 7 ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ)) ∈ V
2523, 24op1st 7980 . . . . . 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 2789 . . . . 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 5861 . . . 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 2799 . . 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 8005 . . . . . 6 (𝐹 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΉ) ∈ 𝐸)
30 xp2nd 8005 . . . . . 6 (𝐺 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΊ) ∈ 𝐸)
31 xp2nd 8005 . . . . . 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 2733 . . . . . . . . . 10 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
342, 33, 5, 6dvhsca 39942 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
352, 33erngdv 39853 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
3634, 35eqeltrd 2834 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
37 drnggrp 20318 . . . . . . . 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 2733 . . . . . . . . 9 (Baseβ€˜π·) = (Baseβ€˜π·)
422, 4, 5, 6, 41dvhbase 39943 . . . . . . . 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 2837 . . . . . 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 2837 . . . . . 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 2837 . . . . . 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 18825 . . . . . 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 6893 . . . . . 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 7981 . . . . . 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 2789 . . . . 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 7421 . . . 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 6893 . . . . . 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 7981 . . . . . 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 2789 . . . . 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 7422 . . . 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 2783 . . 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 4881 . 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 39955 . . . 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 39952 . . 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 39955 . . . 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 39952 . . 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 2783 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 4634   Γ— cxp 5674   ∘ ccom 5680  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  Basecbs 17141  +gcplusg 17194  Scalarcsca 17197  Grpcgrp 18816  DivRingcdr 20308  HLchlt 38209  LHypclh 38844  LTrncltrn 38961  TEndoctendo 39612  EDRingcedring 39613  DVecHcdvh 39938
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 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-riotaBAD 37812
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-tpos 8208  df-undef 8255  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-0g 17384  df-proset 18245  df-poset 18263  df-plt 18280  df-lub 18296  df-glb 18297  df-join 18298  df-meet 18299  df-p0 18375  df-p1 18376  df-lat 18382  df-clat 18449  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-grp 18819  df-minusg 18820  df-mgp 19983  df-ur 20000  df-ring 20052  df-oppr 20143  df-dvdsr 20164  df-unit 20165  df-invr 20195  df-dvr 20208  df-drng 20310  df-oposet 38035  df-ol 38037  df-oml 38038  df-covers 38125  df-ats 38126  df-atl 38157  df-cvlat 38181  df-hlat 38210  df-llines 38358  df-lplanes 38359  df-lvols 38360  df-lines 38361  df-psubsp 38363  df-pmap 38364  df-padd 38656  df-lhyp 38848  df-laut 38849  df-ldil 38964  df-ltrn 38965  df-trl 39019  df-tendo 39615  df-edring 39617  df-dvech 39939
This theorem is referenced by:  dvhgrp  39967
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