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Theorem grpsubsub4 19064
Description: Double group subtraction (subsub4 11540 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 482 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 grpsubadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 grpsubadd.m . . . . . . . 8 = (-g𝐺)
42, 3grpsubcl 19051 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
543adant3r3 1183 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
6 simpr3 1195 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
7 grpsubadd.p . . . . . . 7 + = (+g𝐺)
82, 7, 3grpnpcan 19063 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
91, 5, 6, 8syl3anc 1370 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + 𝑍) = (𝑋 𝑌))
109oveq1d 7446 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = ((𝑋 𝑌) + 𝑌))
112, 3grpsubcl 19051 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
121, 5, 6, 11syl3anc 1370 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
13 simpr2 1194 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
142, 7grpass 18973 . . . . 5 ((𝐺 ∈ Grp ∧ (((𝑋 𝑌) 𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
151, 12, 6, 13, 14syl13anc 1371 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) 𝑍) + 𝑍) + 𝑌) = (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)))
162, 7, 3grpnpcan 19063 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) + 𝑌) = 𝑋)
17163adant3r3 1183 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑌) = 𝑋)
1810, 15, 173eqtr3d 2783 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋)
19 simpr1 1193 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
202, 7grpcl 18972 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 + 𝑌) ∈ 𝐵)
211, 6, 13, 20syl3anc 1370 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 + 𝑌) ∈ 𝐵)
222, 7, 3grpsubadd 19059 . . . 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 1371 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍) ↔ (((𝑋 𝑌) 𝑍) + (𝑍 + 𝑌)) = 𝑋))
2418, 23mpbird 257 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑍 + 𝑌)) = ((𝑋 𝑌) 𝑍))
2524eqcomd 2741 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑍 + 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969
This theorem is referenced by:  grppnpcan2  19065  grpnnncan2  19068  sylow3lem1  19660  subgdisj1  19724  pjthlem2  25486  ply1divex  26191
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