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Theorem nvm 30523
Description: Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvm.1 𝑋 = (BaseSet‘𝑈)
nvm.2 𝐺 = ( +𝑣𝑈)
nvm.3 𝑀 = ( −𝑣𝑈)
nvm.6 𝑁 = ( /𝑔𝐺)
Assertion
Ref Expression
nvm ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))

Proof of Theorem nvm
StepHypRef Expression
1 nvm.2 . . . . 5 𝐺 = ( +𝑣𝑈)
2 nvm.3 . . . . 5 𝑀 = ( −𝑣𝑈)
31, 2vsfval 30515 . . . 4 𝑀 = ( /𝑔𝐺)
4 nvm.6 . . . 4 𝑁 = ( /𝑔𝐺)
53, 4eqtr4i 2756 . . 3 𝑀 = 𝑁
65oveqi 7432 . 2 (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
76a1i 11 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419   /𝑔 cgs 30374  NrmCVeccnv 30466   +𝑣 cpv 30467  BaseSetcba 30468  𝑣 cnsb 30471
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-grpo 30375  df-gdiv 30378  df-va 30477  df-vs 30481
This theorem is referenced by:  nvmval  30524
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