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Theorem ablnnncan1 18936
Description: Cancellation law for group subtraction. (nnncan1 10914 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 18903 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
73, 6syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablsub32.z . . . 4 (𝜑𝑍𝐵)
91, 2grpsubcl 18171 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
107, 4, 8, 9syl3anc 1366 . . 3 (𝜑 → (𝑋 𝑍) ∈ 𝐵)
111, 2, 3, 4, 5, 10ablsub32 18934 . 2 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = ((𝑋 (𝑋 𝑍)) 𝑌))
121, 2, 3, 4, 8ablnncan 18933 . . 3 (𝜑 → (𝑋 (𝑋 𝑍)) = 𝑍)
1312oveq1d 7163 . 2 (𝜑 → ((𝑋 (𝑋 𝑍)) 𝑌) = (𝑍 𝑌))
1411, 13eqtrd 2854 1 (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  cfv 6348  (class class class)co 7148  Basecbs 16475  Grpcgrp 18095  -gcsg 18097  Abelcabl 18899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-cmn 18900  df-abl 18901
This theorem is referenced by:  minveclem2  24021  ply1divmo  24721  baerlem3lem2  38838
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