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Theorem nvadd32 30380
Description: Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvgcl.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nvgcl.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
Assertion
Ref Expression
nvadd32 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺𝐢) = ((𝐴𝐺𝐢)𝐺𝐡))

Proof of Theorem nvadd32
StepHypRef Expression
1 nvgcl.2 . . 3 𝐺 = ( +𝑣 β€˜π‘ˆ)
21nvablo 30373 . 2 (π‘ˆ ∈ NrmCVec β†’ 𝐺 ∈ AbelOp)
3 nvgcl.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
43, 1bafval 30361 . . 3 𝑋 = ran 𝐺
54ablo32 30306 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺𝐢) = ((𝐴𝐺𝐢)𝐺𝐡))
62, 5sylan 579 1 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺𝐢) = ((𝐴𝐺𝐢)𝐺𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  (class class class)co 7404  AbelOpcablo 30301  NrmCVeccnv 30341   +𝑣 cpv 30342  BaseSetcba 30343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-1st 7971  df-2nd 7972  df-grpo 30250  df-ablo 30302  df-vc 30316  df-nv 30349  df-va 30352  df-ba 30353  df-sm 30354  df-0v 30355  df-nmcv 30357
This theorem is referenced by:  nvpncan2  30410
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