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Theorem grpsubadd 18840
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 2733 . . . . . . 7 (invg𝐺) = (invg𝐺)
4 grpsubadd.m . . . . . . 7 = (-g𝐺)
51, 2, 3, 4grpsubval 18801 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant3 1133 . . . . 5 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
76adantl 483 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
87eqeq1d 2735 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
9 simpl 484 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr1 1195 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 3grpinvcl 18803 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
12113ad2antr2 1190 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑌) ∈ 𝐵)
131, 2grpcl 18761 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
149, 10, 12, 13syl3anc 1372 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
15 simpr3 1197 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
16 simpr2 1196 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
171, 2grprcan 18789 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
189, 14, 15, 16, 17syl13anc 1373 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
191, 2grpass 18762 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
209, 10, 12, 16, 19syl13anc 1373 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
21 eqid 2733 . . . . . . . 8 (0g𝐺) = (0g𝐺)
221, 2, 21, 3grplinv 18805 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
23223ad2antr2 1190 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
2423oveq2d 7374 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
251, 2, 21grprid 18786 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
26253ad2antr1 1189 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (0g𝐺)) = 𝑋)
2720, 24, 263eqtrd 2777 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = 𝑋)
2827eqeq1d 2735 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ 𝑋 = (𝑍 + 𝑌)))
298, 18, 283bitr2d 307 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍𝑋 = (𝑍 + 𝑌)))
30 eqcom 2740 . 2 (𝑋 = (𝑍 + 𝑌) ↔ (𝑍 + 𝑌) = 𝑋)
3129, 30bitrdi 287 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753  invgcminusg 18754  -gcsg 18755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758
This theorem is referenced by:  grpsubsub4  18845  conjghm  19044  conjnmzb  19048  sylow3lem2  19415  ablsubadd  19595  ablsubsub23  19608  pgpfac1lem2  19859  pgpfac1lem4  19862  lspexch  20606  ipsubdir  21062  ipsubdi  21063  coe1subfv  21653  lindsunlem  32376  zlmodzxzsub  46522
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