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Theorem grpinvval 18796
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 7366 . . . 4 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
21eqeq1d 2735 . . 3 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
32riotabidv 7316 . 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 18794 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
9 riotaex 7318 . 2 (𝑦𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
103, 8, 9fvmpt 6949 1 (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6497  crio 7313  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  invgcminusg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-minusg 18757
This theorem is referenced by:  grplinv  18805  isgrpinv  18809  xrsinvgval  31917  ringinvval  32119  ressply1invg  32328
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