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Theorem grpinvval 19015
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 7453 . . . 4 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
21eqeq1d 2736 . . 3 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
32riotabidv 7403 . 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 19013 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
9 riotaex 7405 . 2 (𝑦𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
103, 8, 9fvmpt 7027 1 (𝑋𝐵 → (𝑁𝑋) = (𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  cfv 6572  crio 7400  (class class class)co 7445  Basecbs 17253  +gcplusg 17306  0gc0g 17494  invgcminusg 18969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-riota 7401  df-ov 7448  df-minusg 18972
This theorem is referenced by:  grplinv  19024  isgrpinv  19028  xrsinvgval  32983  ringinvval  33207  ressply1invg  33551  linvh  42001  primrootsunit1  42002
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