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Theorem grpinvval 17455
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinvval.b 𝐵 = (Base‘𝐺)
grpinvval.p + = (+g𝐺)
grpinvval.o 0 = (0g𝐺)
grpinvval.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvval (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   + (𝑦)   𝑁(𝑦)   0 (𝑦)

Proof of Theorem grpinvval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6655 . . . 4 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
21eqeq1d 2623 . . 3 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
32riotabidv 6610 . 2 (𝑥 = 𝑋 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
4 grpinvval.b . . 3 𝐵 = (Base‘𝐺)
5 grpinvval.p . . 3 + = (+g𝐺)
6 grpinvval.o . . 3 0 = (0g𝐺)
7 grpinvval.n . . 3 𝑁 = (invg𝐺)
84, 5, 6, 7grpinvfval 17454 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
9 riotaex 6612 . 2 (𝑦𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
103, 8, 9fvmpt 6280 1 (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cfv 5886  crio 6607  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  0gc0g 16094  invgcminusg 17417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-minusg 17420
This theorem is referenced by:  grplinv  17462  isgrpinv  17466  xrsinvgval  29662  ringinvval  29777
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