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Theorem nvm 28047
 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 28039 . . . 4 𝑀 = ( /𝑔𝐺)
4 nvm.6 . . . 4 𝑁 = ( /𝑔𝐺)
53, 4eqtr4i 2852 . . 3 𝑀 = 𝑁
65oveqi 6923 . 2 (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
76a1i 11 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ‘cfv 6127  (class class class)co 6910   /𝑔 cgs 27898  NrmCVeccnv 27990   +𝑣 cpv 27991  BaseSetcba 27992   −𝑣 cnsb 27995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-grpo 27899  df-gdiv 27902  df-va 28001  df-vs 28005 This theorem is referenced by:  nvmval  28048
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