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Theorem nvm 28422
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 28414 . . . 4 𝑀 = ( /𝑔𝐺)
4 nvm.6 . . . 4 𝑁 = ( /𝑔𝐺)
53, 4eqtr4i 2848 . . 3 𝑀 = 𝑁
65oveqi 7153 . 2 (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
76a1i 11 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140   /𝑔 cgs 28273  NrmCVeccnv 28365   +𝑣 cpv 28366  BaseSetcba 28367  𝑣 cnsb 28370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-grpo 28274  df-gdiv 28277  df-va 28376  df-vs 28380
This theorem is referenced by:  nvmval  28423
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