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Theorem nvm 28345
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 28337 . . . 4 𝑀 = ( /𝑔𝐺)
4 nvm.6 . . . 4 𝑁 = ( /𝑔𝐺)
53, 4eqtr4i 2844 . . 3 𝑀 = 𝑁
65oveqi 7158 . 2 (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
76a1i 11 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145   /𝑔 cgs 28196  NrmCVeccnv 28288   +𝑣 cpv 28289  BaseSetcba 28290  𝑣 cnsb 28293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gdiv 28200  df-va 28299  df-vs 28303
This theorem is referenced by:  nvmval  28346
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