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Theorem grppnpcan2 17278
Description: Cancellation law for mixed addition and subtraction. (pnpcan2 10172 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grppnpcan2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) (𝑌 + 𝑍)) = (𝑋 𝑌))

Proof of Theorem grppnpcan2
StepHypRef Expression
1 simpl 471 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
2 grpsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpsubadd.p . . . . 5 + = (+g𝐺)
42, 3grpcl 17199 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
543adant3r2 1266 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑍) ∈ 𝐵)
6 simpr3 1061 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
7 simpr2 1060 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
8 grpsubadd.m . . . 4 = (-g𝐺)
92, 3, 8grpsubsub4 17277 . . 3 ((𝐺 ∈ Grp ∧ ((𝑋 + 𝑍) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + 𝑍) 𝑍) 𝑌) = ((𝑋 + 𝑍) (𝑌 + 𝑍)))
101, 5, 6, 7, 9syl13anc 1319 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + 𝑍) 𝑍) 𝑌) = ((𝑋 + 𝑍) (𝑌 + 𝑍)))
112, 3, 8grppncan 17275 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → ((𝑋 + 𝑍) 𝑍) = 𝑋)
12113adant3r2 1266 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) 𝑍) = 𝑋)
1312oveq1d 6542 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + 𝑍) 𝑍) 𝑌) = (𝑋 𝑌))
1410, 13eqtr3d 2645 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) (𝑌 + 𝑍)) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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 2033  ax-13 2233  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:  ngprcan  22164
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