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Theorem grpsubsub4 17331
 Description: Double group subtraction (subsub4 10193 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
Assertion
Ref Expression
grpsubsub4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))

Proof of Theorem grpsubsub4
StepHypRef Expression
1 simpl 472 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 grpsubadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 grpsubadd.m . . . . . . . 8 = (-g𝐺)
42, 3grpsubcl 17318 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
543adant3r3 1268 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
6 simpr3 1062 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
7 grpsubadd.p . . . . . . 7 + = (+g𝐺)
82, 7, 3grpnpcan 17330 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
91, 5, 6, 8syl3anc 1318 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
109oveq1d 6564 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = ((𝑋 𝑌) + 𝑌))
112, 3grpsubcl 17318 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
121, 5, 6, 11syl3anc 1318 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
13 simpr2 1061 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
142, 7grpass 17254 . . . . 5 ((𝐺 ∈ Grp ∧ (((𝑋 𝑌) 𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
151, 12, 6, 13, 14syl13anc 1320 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
162, 7, 3grpnpcan 17330 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
17163adant3r3 1268 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑌) = 𝑋)
1810, 15, 173eqtr3d 2652 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋)
19 simpr1 1060 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
202, 7grpcl 17253 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
211, 6, 13, 20syl3anc 1318 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
222, 7, 3grpsubadd 17326 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑍 + 𝑌) ∈ 𝐵 ∧ ((𝑋 𝑌) 𝑍) ∈ 𝐵)) → ((𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍) ↔ (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋))
231, 19, 21, 12, 22syl13anc 1320 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍) ↔ (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋))
2418, 23mpbird 246 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍))
2524eqcomd 2616 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  -gcsg 17247 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250 This theorem is referenced by:  grppnpcan2  17332  grpnnncan2  17335  sylow3lem1  17865  subgdisj1  17927  pjthlem2  23017  ply1divex  23700
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