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Theorem grpinvval2 18185
Description: A df-neg 10872-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 18134 . . 3 (𝐺 ∈ Grp → 0𝐵)
4 eqid 2824 . . . 4 (+g𝐺) = (+g𝐺)
5 grpinvsub.n . . . 4 𝑁 = (invg𝐺)
6 grpsubcl.m . . . 4 = (-g𝐺)
71, 4, 5, 6grpsubval 18152 . . 3 (( 0𝐵𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
83, 7sylan 583 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 𝑋) = ( 0 (+g𝐺)(𝑁𝑋)))
91, 5grpinvcl 18154 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
101, 4, 2grplid 18136 . . 3 ((𝐺 ∈ Grp ∧ (𝑁𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
119, 10syldan 594 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 (+g𝐺)(𝑁𝑋)) = (𝑁𝑋))
128, 11eqtr2d 2860 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) = ( 0 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cfv 6344  (class class class)co 7150  Basecbs 16486  +gcplusg 16568  0gc0g 16716  Grpcgrp 18106  invgcminusg 18107  -gcsg 18108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-0g 16718  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-grp 18109  df-minusg 18110  df-sbg 18111
This theorem is referenced by:  grpsubadd0sub  18189  matinvgcell  21047  istgp2  22702  nrmmetd  23187  nminv  23233
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