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Theorem nvm 27466
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 27458 . . . 4 𝑀 = ( /𝑔𝐺)
4 nvm.6 . . . 4 𝑁 = ( /𝑔𝐺)
53, 4eqtr4i 2645 . . 3 𝑀 = 𝑁
65oveqi 6648 . 2 (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
76a1i 11 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635   /𝑔 cgs 27316  NrmCVeccnv 27409   +𝑣 cpv 27410  BaseSetcba 27411  𝑣 cnsb 27414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-grpo 27317  df-gdiv 27320  df-va 27420  df-vs 27424
This theorem is referenced by:  nvmval  27467
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