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Theorem vsfval 27839
Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
vsfval.2 𝐺 = ( +𝑣𝑈)
vsfval.3 𝑀 = ( −𝑣𝑈)
Assertion
Ref Expression
vsfval 𝑀 = ( /𝑔𝐺)

Proof of Theorem vsfval
Dummy variables 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vs 27805 . . . . 5 𝑣 = ( /𝑔 ∘ +𝑣 )
21fveq1i 6419 . . . 4 ( −𝑣𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3 fo1st 7428 . . . . . . . 8 1st :V–onto→V
4 fof 6341 . . . . . . . 8 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . . . . 7 1st :V⟶V
6 fco 6283 . . . . . . 7 ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
75, 5, 6mp2an 675 . . . . . 6 (1st ∘ 1st ):V⟶V
8 df-va 27801 . . . . . . 7 +𝑣 = (1st ∘ 1st )
98feq1i 6257 . . . . . 6 ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
107, 9mpbir 222 . . . . 5 +𝑣 :V⟶V
11 fvco3 6506 . . . . 5 (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
1210, 11mpan 673 . . . 4 (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
132, 12syl5eq 2863 . . 3 (𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
14 0ngrp 27717 . . . . . 6 ¬ ∅ ∈ GrpOp
15 vex 3405 . . . . . . . . . 10 𝑔 ∈ V
1615rnex 7340 . . . . . . . . 9 ran 𝑔 ∈ V
1716, 16mpt2ex 7490 . . . . . . . 8 (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18 df-gdiv 27702 . . . . . . . 8 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1917, 18dmmpti 6244 . . . . . . 7 dom /𝑔 = GrpOp
2019eleq2i 2888 . . . . . 6 (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
2114, 20mtbir 314 . . . . 5 ¬ ∅ ∈ dom /𝑔
22 ndmfv 6448 . . . . 5 (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
2321, 22mp1i 13 . . . 4 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24 fvprc 6411 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
2524fveq2d 6422 . . . 4 𝑈 ∈ V → ( /𝑔 ‘( +𝑣𝑈)) = ( /𝑔 ‘∅))
26 fvprc 6411 . . . 4 𝑈 ∈ V → ( −𝑣𝑈) = ∅)
2723, 25, 263eqtr4rd 2862 . . 3 𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
2813, 27pm2.61i 176 . 2 ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈))
29 vsfval.3 . 2 𝑀 = ( −𝑣𝑈)
30 vsfval.2 . . 3 𝐺 = ( +𝑣𝑈)
3130fveq2i 6421 . 2 ( /𝑔𝐺) = ( /𝑔 ‘( +𝑣𝑈))
3228, 29, 313eqtr4i 2849 1 𝑀 = ( /𝑔𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1637  wcel 2157  Vcvv 3402  c0 4127  dom cdm 5324  ran crn 5325  ccom 5328  wf 6107  ontowfo 6109  cfv 6111  (class class class)co 6884  cmpt2 6886  1st c1st 7406  GrpOpcgr 27695  invcgn 27697   /𝑔 cgs 27698   +𝑣 cpv 27791  𝑣 cnsb 27795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-1st 7408  df-2nd 7409  df-grpo 27699  df-gdiv 27702  df-va 27801  df-vs 27805
This theorem is referenced by:  nvm  27847  nvmfval  27850  nvnnncan1  27853  nvaddsub  27861
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