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Theorem grpsubsub4 17277
Description: Double group subtraction (subsub4 10165 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpsubsub4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))

Proof of Theorem grpsubsub4
StepHypRef Expression
1 simpl 471 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 grpsubadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 grpsubadd.m . . . . . . . 8 = (-g𝐺)
42, 3grpsubcl 17264 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
543adant3r3 1267 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
6 simpr3 1061 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
7 grpsubadd.p . . . . . . 7 + = (+g𝐺)
82, 7, 3grpnpcan 17276 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
91, 5, 6, 8syl3anc 1317 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
109oveq1d 6542 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = ((𝑋 𝑌) + 𝑌))
112, 3grpsubcl 17264 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
121, 5, 6, 11syl3anc 1317 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
13 simpr2 1060 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
142, 7grpass 17200 . . . . 5 ((𝐺 ∈ Grp ∧ (((𝑋 𝑌) 𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
151, 12, 6, 13, 14syl13anc 1319 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
162, 7, 3grpnpcan 17276 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
17163adant3r3 1267 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑌) = 𝑋)
1810, 15, 173eqtr3d 2651 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋)
19 simpr1 1059 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
202, 7grpcl 17199 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
211, 6, 13, 20syl3anc 1317 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
222, 7, 3grpsubadd 17272 . . . 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 1319 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍) ↔ (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋))
2418, 23mpbird 245 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍))
2524eqcomd 2615 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Grpcgrp 17191  -gcsg 17193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-minusg 17195  df-sbg 17196
This theorem is referenced by:  grppnpcan2  17278  grpnnncan2  17281  sylow3lem1  17811  subgdisj1  17873  pjthlem2  22934  ply1divex  23617
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