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Theorem grpsubadd 17944
Description: Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpsubadd ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))

Proof of Theorem grpsubadd
StepHypRef Expression
1 grpsubadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
2 grpsubadd.p . . . . . . 7 + = (+g𝐺)
3 eqid 2794 . . . . . . 7 (invg𝐺) = (invg𝐺)
4 grpsubadd.m . . . . . . 7 = (-g𝐺)
51, 2, 3, 4grpsubval 17906 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant3 1125 . . . . 5 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
76adantl 482 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
87eqeq1d 2796 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
9 simpl 483 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr1 1187 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 3grpinvcl 17908 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
12113ad2antr2 1182 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑌) ∈ 𝐵)
131, 2grpcl 17869 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
149, 10, 12, 13syl3anc 1364 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
15 simpr3 1189 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
16 simpr2 1188 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
171, 2grprcan 17894 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
189, 14, 15, 16, 17syl13anc 1365 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
191, 2grpass 17870 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
209, 10, 12, 16, 19syl13anc 1365 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
21 eqid 2794 . . . . . . . 8 (0g𝐺) = (0g𝐺)
221, 2, 21, 3grplinv 17909 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
23223ad2antr2 1182 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
2423oveq2d 7035 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
251, 2, 21grprid 17892 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
26253ad2antr1 1181 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (0g𝐺)) = 𝑋)
2720, 24, 263eqtrd 2834 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = 𝑋)
2827eqeq1d 2796 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ 𝑋 = (𝑍 + 𝑌)))
298, 18, 283bitr2d 308 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍𝑋 = (𝑍 + 𝑌)))
30 eqcom 2801 . 2 (𝑋 = (𝑍 + 𝑌) ↔ (𝑍 + 𝑌) = 𝑋)
3129, 30syl6bb 288 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2080  cfv 6228  (class class class)co 7019  Basecbs 16312  +gcplusg 16394  0gc0g 16542  Grpcgrp 17861  invgcminusg 17862  -gcsg 17863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866
This theorem is referenced by:  grpsubsub4  17949  conjghm  18130  conjnmzb  18134  sylow3lem2  18483  ablsubadd  18657  ablsubsub23  18670  pgpfac1lem2  18914  pgpfac1lem4  18917  lspexch  19591  coe1subfv  20117  ipsubdir  20468  ipsubdi  20469  lindsunlem  30616  zlmodzxzsub  43900
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