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Theorem grpsubsub4 17777
Description: Double group subtraction (subsub4 10568 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 474 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 grpsubadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 grpsubadd.m . . . . . . . 8 = (-g𝐺)
42, 3grpsubcl 17764 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
543adant3r3 1235 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
6 simpr3 1252 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
7 grpsubadd.p . . . . . . 7 + = (+g𝐺)
82, 7, 3grpnpcan 17776 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
91, 5, 6, 8syl3anc 1490 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
109oveq1d 6857 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = ((𝑋 𝑌) + 𝑌))
112, 3grpsubcl 17764 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
121, 5, 6, 11syl3anc 1490 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
13 simpr2 1250 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
142, 7grpass 17700 . . . . 5 ((𝐺 ∈ Grp ∧ (((𝑋 𝑌) 𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
151, 12, 6, 13, 14syl13anc 1491 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
162, 7, 3grpnpcan 17776 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
17163adant3r3 1235 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑌) = 𝑋)
1810, 15, 173eqtr3d 2807 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋)
19 simpr1 1248 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
202, 7grpcl 17699 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
211, 6, 13, 20syl3anc 1490 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
222, 7, 3grpsubadd 17772 . . . 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 1491 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍) ↔ (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋))
2418, 23mpbird 248 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍))
2524eqcomd 2771 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Grpcgrp 17691  -gcsg 17693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-0g 16370  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-grp 17694  df-minusg 17695  df-sbg 17696
This theorem is referenced by:  grppnpcan2  17778  grpnnncan2  17781  sylow3lem1  18308  subgdisj1  18370  pjthlem2  23498  ply1divex  24187
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