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Theorem dvhvaddass 40271
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 6263 . . . 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 40266 . . . . . . . 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 1169 . . . . . . 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 6894 . . . . . 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 6903 . . . . . . . 8 (1st β€˜πΉ) ∈ V
13 fvex 6903 . . . . . . . 8 (1st β€˜πΊ) ∈ V
1412, 13coex 7923 . . . . . . 7 ((1st β€˜πΉ) ∘ (1st β€˜πΊ)) ∈ V
15 ovex 7444 . . . . . . 7 ((2nd β€˜πΉ) ⨣ (2nd β€˜πΊ)) ∈ V
1614, 15op1st 7985 . . . . . 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 2786 . . . . 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 40266 . . . . . . . 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 1167 . . . . . . 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 6894 . . . . . 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 6903 . . . . . . . 8 (1st β€˜πΌ) ∈ V
2313, 22coex 7923 . . . . . . 7 ((1st β€˜πΊ) ∘ (1st β€˜πΌ)) ∈ V
24 ovex 7444 . . . . . . 7 ((2nd β€˜πΊ) ⨣ (2nd β€˜πΌ)) ∈ V
2523, 24op1st 7985 . . . . . 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 2786 . . . . 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 2796 . . 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 8010 . . . . . 6 (𝐹 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΉ) ∈ 𝐸)
30 xp2nd 8010 . . . . . 6 (𝐺 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΊ) ∈ 𝐸)
31 xp2nd 8010 . . . . . 6 (𝐼 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜πΌ) ∈ 𝐸)
3229, 30, 313anim123i 1149 . . . . 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 2730 . . . . . . . . . 10 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
342, 33, 5, 6dvhsca 40256 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
352, 33erngdv 40167 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
3634, 35eqeltrd 2831 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
37 drnggrp 20510 . . . . . . . 8 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Grp)
3836, 37syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
3938adantr 479 . . . . . 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 1192 . . . . . . 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 2730 . . . . . . . . 9 (Baseβ€˜π·) = (Baseβ€˜π·)
422, 4, 5, 6, 41dvhbase 40257 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
4342adantr 479 . . . . . . 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 2834 . . . . . 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 1193 . . . . . . 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 2834 . . . . . 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 1194 . . . . . . 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 2834 . . . . . 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 18864 . . . . . 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 1370 . . . . 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 591 . . . 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 6894 . . . . . 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 7986 . . . . . 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 2786 . . . . 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 7426 . . . 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 6894 . . . . . 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 7986 . . . . . 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 2786 . . . . 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 7427 . . . 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 2780 . . 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 4880 . 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 481 . . 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 40269 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 Γ— 𝐸) ∧ 𝐺 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐹 + 𝐺) ∈ (𝑇 Γ— 𝐸))
64633adantr3 1169 . . 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 1194 . . 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 40266 . . 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 833 . 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 1192 . . 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 40269 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 Γ— 𝐸) ∧ 𝐼 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐺 + 𝐼) ∈ (𝑇 Γ— 𝐸))
70693adantr1 1167 . . 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 40266 . . 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 833 . 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 2780 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βŸ¨cop 4633   Γ— cxp 5673   ∘ ccom 5679  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  Basecbs 17148  +gcplusg 17201  Scalarcsca 17204  Grpcgrp 18855  DivRingcdr 20500  HLchlt 38523  LHypclh 39158  LTrncltrn 39275  TEndoctendo 39926  EDRingcedring 39927  DVecHcdvh 40252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-riotaBAD 38126
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-undef 8260  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-0g 17391  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-dvr 20292  df-drng 20502  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-llines 38672  df-lplanes 38673  df-lvols 38674  df-lines 38675  df-psubsp 38677  df-pmap 38678  df-padd 38970  df-lhyp 39162  df-laut 39163  df-ldil 39278  df-ltrn 39279  df-trl 39333  df-tendo 39929  df-edring 39931  df-dvech 40253
This theorem is referenced by:  dvhgrp  40281
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