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Theorem grpsubadd 18988
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 2725 . . . . . . 7 (invg𝐺) = (invg𝐺)
4 grpsubadd.m . . . . . . 7 = (-g𝐺)
51, 2, 3, 4grpsubval 18946 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant3 1129 . . . . 5 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
76adantl 480 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
87eqeq1d 2727 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
9 simpl 481 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr1 1191 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 3grpinvcl 18948 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
12113ad2antr2 1186 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑌) ∈ 𝐵)
131, 2grpcl 18902 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
149, 10, 12, 13syl3anc 1368 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
15 simpr3 1193 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
16 simpr2 1192 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
171, 2grprcan 18934 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
189, 14, 15, 16, 17syl13anc 1369 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
191, 2grpass 18903 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
209, 10, 12, 16, 19syl13anc 1369 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
21 eqid 2725 . . . . . . . 8 (0g𝐺) = (0g𝐺)
221, 2, 21, 3grplinv 18950 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
23223ad2antr2 1186 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
2423oveq2d 7432 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
251, 2, 21grprid 18929 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
26253ad2antr1 1185 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (0g𝐺)) = 𝑋)
2720, 24, 263eqtrd 2769 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = 𝑋)
2827eqeq1d 2727 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ 𝑋 = (𝑍 + 𝑌)))
298, 18, 283bitr2d 306 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍𝑋 = (𝑍 + 𝑌)))
30 eqcom 2732 . 2 (𝑋 = (𝑍 + 𝑌) ↔ (𝑍 + 𝑌) = 𝑋)
3129, 30bitrdi 286 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cfv 6543  (class class class)co 7416  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18894  invgcminusg 18895  -gcsg 18896
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 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  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 5144  df-opab 5206  df-mpt 5227  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-minusg 18898  df-sbg 18899
This theorem is referenced by:  grpsubsub4  18993  xpsgrpsub  19021  conjghm  19207  conjnmzb  19211  sylow3lem2  19587  ablsubadd  19768  ablsubsub23  19783  pgpfac1lem2  20036  pgpfac1lem4  20039  lspexch  21021  ipsubdir  21578  ipsubdi  21579  coe1subfv  22194  lindsunlem  33379  zlmodzxzsub  47536
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