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Theorem ablnnncan1 19743
Description: Cancellation law for group subtraction. (nnncan1 11500 analog.) (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablnncan.b 𝐵 = (Base‘𝐺)
ablnncan.m = (-g𝐺)
ablnncan.g (𝜑𝐺 ∈ Abel)
ablnncan.x (𝜑𝑋𝐵)
ablnncan.y (𝜑𝑌𝐵)
ablsub32.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablnnncan1 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))

Proof of Theorem ablnnncan1
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 ablnncan.m . . 3 = (-g𝐺)
3 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
4 ablnncan.x . . 3 (𝜑𝑋𝐵)
5 ablnncan.y . . 3 (𝜑𝑌𝐵)
6 ablgrp 19705 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
73, 6syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablsub32.z . . . 4 (𝜑𝑍𝐵)
91, 2grpsubcl 18948 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
107, 4, 8, 9syl3anc 1368 . . 3 (𝜑 → (𝑋 𝑍) ∈ 𝐵)
111, 2, 3, 4, 5, 10ablsub32 19741 . 2 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = ((𝑋 (𝑋 𝑍)) 𝑌))
121, 2, 3, 4, 8ablnncan 19740 . . 3 (𝜑 → (𝑋 (𝑋 𝑍)) = 𝑍)
1312oveq1d 7420 . 2 (𝜑 → ((𝑋 (𝑋 𝑍)) 𝑌) = (𝑍 𝑌))
1411, 13eqtrd 2766 1 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  Grpcgrp 18863  -gcsg 18865  Abelcabl 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-cmn 19702  df-abl 19703
This theorem is referenced by:  minveclem2  25309  ply1divmo  26026  baerlem3lem2  41094
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