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Theorem nvnnncan1 29005
Description: Cancellation law for vector subtraction. (nnncan1 11257 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmf.1 𝑋 = (BaseSet‘𝑈)
nvmf.3 𝑀 = ( −𝑣𝑈)
Assertion
Ref Expression
nvnnncan1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))

Proof of Theorem nvnnncan1
StepHypRef Expression
1 eqid 2740 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
21nvablo 28974 . 2 (𝑈 ∈ NrmCVec → ( +𝑣𝑈) ∈ AbelOp)
3 nvmf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
43, 1bafval 28962 . . 3 𝑋 = ran ( +𝑣𝑈)
5 nvmf.3 . . . 4 𝑀 = ( −𝑣𝑈)
61, 5vsfval 28991 . . 3 𝑀 = ( /𝑔 ‘( +𝑣𝑈))
74, 6ablonnncan1 28915 . 2 ((( +𝑣𝑈) ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))
82, 7sylan 580 1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  cfv 6432  (class class class)co 7271  AbelOpcablo 28902  NrmCVeccnv 28942   +𝑣 cpv 28943  BaseSetcba 28944  𝑣 cnsb 28947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-grpo 28851  df-gid 28852  df-ginv 28853  df-gdiv 28854  df-ablo 28903  df-vc 28917  df-nv 28950  df-va 28953  df-ba 28954  df-sm 28955  df-0v 28956  df-vs 28957  df-nmcv 28958
This theorem is referenced by:  minvecolem2  29233
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