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Theorem ablnnncan 19608
Description: Cancellation law for group subtraction. (nnncan 11443 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
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
ablnnncan (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))

Proof of Theorem ablnnncan
StepHypRef Expression
1 ablnncan.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2737 . . 3 (+g𝐺) = (+g𝐺)
3 ablnncan.m . . 3 = (-g𝐺)
4 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
5 ablnncan.x . . 3 (𝜑𝑋𝐵)
6 ablgrp 19574 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
74, 6syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablnncan.y . . . 4 (𝜑𝑌𝐵)
9 ablsub32.z . . . 4 (𝜑𝑍𝐵)
101, 3grpsubcl 18834 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
117, 8, 9, 10syl3anc 1372 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
121, 2, 3, 4, 5, 11, 9ablsubsub4 19604 . 2 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
131, 2ablcom 19588 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑌 𝑍) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
144, 11, 9, 13syl3anc 1372 . . . 4 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
151, 2, 3ablpncan3 19602 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑌𝐵)) → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
164, 9, 8, 15syl12anc 836 . . . 4 (𝜑 → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
1714, 16eqtrd 2777 . . 3 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1817oveq2d 7378 . 2 (𝜑 → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1912, 18eqtrd 2777 1 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  Grpcgrp 18755  -gcsg 18757  Abelcabl 19570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-cmn 19571  df-abl 19572
This theorem is referenced by: (None)
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