Theorem nvadd12 8238

Description: Commutative/associative law for vector addition.

Hypotheses

Ref	Expression
nvgcl.1	⊢ X = (Base ‘U)
nvgcl.2	⊢ G = ( +v ‘U)

Assertion

Ref	Expression
nvadd12	⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → (AG(BGC)) = (BG(AGC)))

Proof of Theorem nvadd12

Step	Hyp	Ref	Expression
1		nvgcl.1	. . . . 5 ⊢ X = (Base ‘U)
2		nvgcl.2	. . . . 5 ⊢ G = ( +v ‘U)
3	1, 2	nvcom 8236	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ A ∈ X ⋀ B ∈ X) → (AGB) = (BGA))
4	3	3adant3r3 846	. . 3 ⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → (AGB) = (BGA))
5	4	opreq1d 3981	. 2 ⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → ((AGB)GC) = ((BGA)GC))
6	1, 2	nvass 8237	. 2 ⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → ((AGB)GC) = (AG(BGC)))
7	1, 2	nvass 8237	. . 3 ⊢ ((U ∈ NrmCVec ⋀ (B ∈ X ⋀ A ∈ X ⋀ C ∈ X)) → ((BGA)GC) = (BG(AGC)))
8		3ancoma 784	. . 3 ⊢ ((A ∈ X ⋀ B ∈ X ⋀ C ∈ X) ↔ (B ∈ X ⋀ A ∈ X ⋀ C ∈ X))
9	7, 8	sylan2b 454	. 2 ⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → ((BGA)GC) = (BG(AGC)))
10	5, 6, 9	3eqtr3d 1518	1 ⊢ ((U ∈ NrmCVec ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → (AG(BGC)) = (BG(AGC)))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960   ‘cfv 3188  (class class class)co 3969  NrmCVeccnv 8199   +_v cpv 8200  Basecba 8201

This theorem is referenced by:  nvsubadd 8271

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215

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