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Theorem grpinvval2 17269
Description: A df-neg 10120-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubcl.b 𝐵 = (Base‘𝐺)
grpsubcl.m = (-g𝐺)
grpinvsub.n 𝑁 = (invg𝐺)
grpinvval2.z 0 = (0g𝐺)
Assertion
Ref Expression
grpinvval2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))

Proof of Theorem grpinvval2
StepHypRef Expression
1 grpsubcl.b . . . 4 𝐵 = (Base‘𝐺)
2 grpinvval2.z . . . 4 0 = (0g𝐺)
31, 2grpidcl 17221 . . 3 (𝐺 ∈ Grp → 0𝐵)
4 eqid 2609 . . . 4 (+g𝐺) = (+g𝐺)
5 grpinvsub.n . . . 4 𝑁 = (invg𝐺)
6 grpsubcl.m . . . 4 = (-g𝐺)
71, 4, 5, 6grpsubval 17236 . . 3 (( 0𝐵𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
83, 7sylan 486 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
91, 5grpinvcl 17238 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
101, 4, 2grplid 17223 . . 3 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
119, 10syldan 485 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
128, 11eqtr2d 2644 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5789  (class class class)co 6526  Basecbs 15643  +gcplusg 15716  0gc0g 15871  Grpcgrp 17193  invgcminusg 17194  -gcsg 17195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-0g 15873  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-grp 17196  df-minusg 17197  df-sbg 17198
This theorem is referenced by:  grpsubadd0sub  17273  matinvgcell  20007  istgp2  21652  nrmmetd  22136  nminv  22182
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