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Theorem dvhvaddass 39968
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 6265 . . . 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 39963 . . . . . . . 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 6896 . . . . . 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 6905 . . . . . . . 8 (1st β€˜πΉ) ∈ V
13 fvex 6905 . . . . . . . 8 (1st β€˜πΊ) ∈ V
1412, 13coex 7921 . . . . . . 7 ((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∈ V
15 ovex 7442 . . . . . . 7 ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ∈ V
1614, 15op1st 7983 . . . . . 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 5862 . . . 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 39963 . . . . . . . 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 6896 . . . . . 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 6905 . . . . . . . 8 (1st β€˜πΌ) ∈ V
2313, 22coex 7921 . . . . . . 7 ((1st β€˜πΊ) ∘ (1st β€˜πΌ)) ∈ V
24 ovex 7442 . . . . . . 7 ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ)) ∈ V
2523, 24op1st 7983 . . . . . 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 5863 . . . 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 8008 . . . . . 6 (𝐹 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΉ) ∈ 𝐸)
30 xp2nd 8008 . . . . . 6 (𝐺 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΊ) ∈ 𝐸)
31 xp2nd 8008 . . . . . 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 39953 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
352, 33erngdv 39864 . . . . . . . . 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 20367 . . . . . . . 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 39954 . . . . . . . 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 18828 . . . . . 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 6896 . . . . . 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 7984 . . . . . 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 7424 . . . 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 6896 . . . . . 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 7984 . . . . . 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 7425 . . . 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 4882 . 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 39966 . . . 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 39963 . . 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 39966 . . . 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 39963 . . 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 4635   Γ— cxp 5675   ∘ ccom 5681  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200  Grpcgrp 18819  DivRingcdr 20357  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  TEndoctendo 39623  EDRingcedring 39624  DVecHcdvh 39949
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-riotaBAD 37823
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-undef 8258  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-0g 17387  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-mgp 19988  df-ur 20005  df-ring 20058  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-dvr 20215  df-drng 20359  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030  df-tendo 39626  df-edring 39628  df-dvech 39950
This theorem is referenced by:  dvhgrp  39978
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