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Theorem ablnncan 18207
Description: Cancellation law for group subtraction. (nncan 10295 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)
Assertion
Ref Expression
ablnncan (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)

Proof of Theorem ablnncan
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2620 . . 3 (+g𝐺) = (+g𝐺)
3 ablnncan.m . . 3 = (-g𝐺)
4 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
5 ablnncan.x . . 3 (𝜑𝑋𝐵)
6 ablnncan.y . . 3 (𝜑𝑌𝐵)
71, 2, 3, 4, 5, 5, 6ablsubsub 18204 . 2 (𝜑 → (𝑋 (𝑋 𝑌)) = ((𝑋 𝑋)(+g𝐺)𝑌))
8 ablgrp 18179 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
94, 8syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
10 eqid 2620 . . . . 5 (0g𝐺) = (0g𝐺)
111, 10, 3grpsubid 17480 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
129, 5, 11syl2anc 692 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1312oveq1d 6650 . 2 (𝜑 → ((𝑋 𝑋)(+g𝐺)𝑌) = ((0g𝐺)(+g𝐺)𝑌))
141, 2, 10grplid 17433 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺)(+g𝐺)𝑌) = 𝑌)
159, 6, 14syl2anc 692 . 2 (𝜑 → ((0g𝐺)(+g𝐺)𝑌) = 𝑌)
167, 13, 153eqtrd 2658 1 (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Grpcgrp 17403  -gcsg 17405  Abelcabl 18175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-sbg 17408  df-cmn 18176  df-abl 18177
This theorem is referenced by:  ablnnncan1  18210  pgpfac1lem3  18457  tsmsxplem1  21937  baerlem5blem2  36820
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