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

StepHypRef Expression
1 grpsubadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
2 grpsubadd.p . . . . . . 7 + = (+g𝐺)
3 eqid 2759 . . . . . . 7 (invg𝐺) = (invg𝐺)
4 grpsubadd.m . . . . . . 7 = (-g𝐺)
51, 2, 3, 4grpsubval 18209 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant3 1130 . . . . 5 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
76adantl 486 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
87eqeq1d 2761 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
9 simpl 487 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr1 1192 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 3grpinvcl 18211 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
12113ad2antr2 1187 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑌) ∈ 𝐵)
131, 2grpcl 18170 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
149, 10, 12, 13syl3anc 1369 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
15 simpr3 1194 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
16 simpr2 1193 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
171, 2grprcan 18197 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
189, 14, 15, 16, 17syl13anc 1370 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
191, 2grpass 18171 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
209, 10, 12, 16, 19syl13anc 1370 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
21 eqid 2759 . . . . . . . 8 (0g𝐺) = (0g𝐺)
221, 2, 21, 3grplinv 18212 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
23223ad2antr2 1187 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
2423oveq2d 7167 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
251, 2, 21grprid 18194 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
26253ad2antr1 1186 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (0g𝐺)) = 𝑋)
2720, 24, 263eqtrd 2798 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = 𝑋)
2827eqeq1d 2761 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ 𝑋 = (𝑍 + 𝑌)))
298, 18, 283bitr2d 311 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍𝑋 = (𝑍 + 𝑌)))
30 eqcom 2766 . 2 (𝑋 = (𝑍 + 𝑌) ↔ (𝑍 + 𝑌) = 𝑋)
3129, 30bitrdi 290 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ‘cfv 6336  (class class class)co 7151  Basecbs 16534  +gcplusg 16616  0gc0g 16764  Grpcgrp 18162  invgcminusg 18163  -gcsg 18164 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-0g 16766  df-mgm 17911  df-sgrp 17960  df-mnd 17971  df-grp 18165  df-minusg 18166  df-sbg 18167 This theorem is referenced by:  grpsubsub4  18252  conjghm  18449  conjnmzb  18453  sylow3lem2  18813  ablsubadd  18993  ablsubsub23  19006  pgpfac1lem2  19258  pgpfac1lem4  19261  lspexch  19962  ipsubdir  20400  ipsubdi  20401  coe1subfv  20983  lindsunlem  31219  zlmodzxzsub  45122
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