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Theorem ablnnncan 13229
Description: Cancellation law for group subtraction. (nnncan 8211 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 2189 . . 3 (+g𝐺) = (+g𝐺)
3 ablnncan.m . . 3 = (-g𝐺)
4 ablnncan.g . . 3 (𝜑𝐺 ∈ Abel)
5 ablnncan.x . . 3 (𝜑𝑋𝐵)
6 ablgrp 13195 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
74, 6syl 14 . . . 4 (𝜑𝐺 ∈ Grp)
8 ablnncan.y . . . 4 (𝜑𝑌𝐵)
9 ablsub32.z . . . 4 (𝜑𝑍𝐵)
101, 3grpsubcl 12996 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
117, 8, 9, 10syl3anc 1249 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
121, 2, 3, 4, 5, 11, 9ablsubsub4 13225 . 2 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)))
131, 2ablcom 13209 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑌 𝑍) ∈ 𝐵𝑍𝐵) → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
144, 11, 9, 13syl3anc 1249 . . . 4 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = (𝑍(+g𝐺)(𝑌 𝑍)))
151, 2, 3ablpncan3 13223 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑍𝐵𝑌𝐵)) → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
164, 9, 8, 15syl12anc 1247 . . . 4 (𝜑 → (𝑍(+g𝐺)(𝑌 𝑍)) = 𝑌)
1714, 16eqtrd 2222 . . 3 (𝜑 → ((𝑌 𝑍)(+g𝐺)𝑍) = 𝑌)
1817oveq2d 5907 . 2 (𝜑 → (𝑋 ((𝑌 𝑍)(+g𝐺)𝑍)) = (𝑋 𝑌))
1912, 18eqtrd 2222 1 (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  cfv 5231  (class class class)co 5891  Basecbs 12486  +gcplusg 12561  Grpcgrp 12917  -gcsg 12919  Abelcabl 13191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-plusg 12574  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921  df-sbg 12922  df-cmn 13192  df-abl 13193
This theorem is referenced by: (None)
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