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Theorem grpsubadd0sub 18982
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 2725 . . . 4 (invg𝐺) = (invg𝐺)
4 grpsubid.m . . . 4 = (-g𝐺)
51, 2, 3, 4grpsubval 18941 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant1 1127 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
7 grpsubid.o . . . . 5 0 = (0g𝐺)
81, 4, 3, 7grpinvval2 18978 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) = ( 0 𝑌))
983adant2 1128 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((invg𝐺)‘𝑌) = ( 0 𝑌))
109oveq2d 7429 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) = (𝑋 + ( 0 𝑌)))
116, 10eqtrd 2765 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ( 0 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cfv 6543  (class class class)co 7413  Basecbs 17174  +gcplusg 17227  0gc0g 17415  Grpcgrp 18889  invgcminusg 18890  -gcsg 18891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-0g 17417  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18892  df-minusg 18893  df-sbg 18894
This theorem is referenced by:  chfacfscmulgsum  22775  chfacfpmmulgsum  22779
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