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Theorem grpsubadd0sub 18967
Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
Hypotheses
Ref Expression
grpsubid.b 𝐵 = (Base‘𝐺)
grpsubid.o 0 = (0g𝐺)
grpsubid.m = (-g𝐺)
grpsubadd0sub.p + = (+g𝐺)
Assertion
Ref Expression
grpsubadd0sub ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ( 0 𝑌)))

Proof of Theorem grpsubadd0sub
StepHypRef Expression
1 grpsubid.b . . . 4 𝐵 = (Base‘𝐺)
2 grpsubadd0sub.p . . . 4 + = (+g𝐺)
3 eqid 2727 . . . 4 (invg𝐺) = (invg𝐺)
4 grpsubid.m . . . 4 = (-g𝐺)
51, 2, 3, 4grpsubval 18927 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant1 1128 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
7 grpsubid.o . . . . 5 0 = (0g𝐺)
81, 4, 3, 7grpinvval2 18963 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) = ( 0 𝑌))
983adant2 1129 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘𝑌) = ( 0 𝑌))
109oveq2d 7430 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) = (𝑋 + ( 0 𝑌)))
116, 10eqtrd 2767 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ( 0 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  Basecbs 17165  +gcplusg 17218  0gc0g 17406  Grpcgrp 18875  invgcminusg 18876  -gcsg 18877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-0g 17408  df-mgm 18585  df-sgrp 18664  df-mnd 18680  df-grp 18878  df-minusg 18879  df-sbg 18880
This theorem is referenced by:  chfacfscmulgsum  22736  chfacfpmmulgsum  22740
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