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Theorem ablsub32 18587
 Description: Swap the second and third terms in a double group subtraction. (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
ablsub32 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Proof of Theorem ablsub32
StepHypRef Expression
1 ablnncan.g . . . 4 (𝜑𝐺 ∈ Abel)
2 ablnncan.y . . . 4 (𝜑𝑌𝐵)
3 ablsub32.z . . . 4 (𝜑𝑍𝐵)
4 ablnncan.b . . . . 5 𝐵 = (Base‘𝐺)
5 eqid 2825 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 18570 . . . 4 ((𝐺 ∈ Abel ∧ 𝑌𝐵𝑍𝐵) → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
71, 2, 3, 6syl3anc 1494 . . 3 (𝜑 → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
87oveq2d 6926 . 2 (𝜑 → (𝑋 (𝑌(+g𝐺)𝑍)) = (𝑋 (𝑍(+g𝐺)𝑌)))
9 ablnncan.m . . 3 = (-g𝐺)
10 ablnncan.x . . 3 (𝜑𝑋𝐵)
114, 5, 9, 1, 10, 2, 3ablsubsub4 18584 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌(+g𝐺)𝑍)))
124, 5, 9, 1, 10, 3, 2ablsubsub4 18584 . 2 (𝜑 → ((𝑋 𝑍) 𝑌) = (𝑋 (𝑍(+g𝐺)𝑌)))
138, 11, 123eqtr4d 2871 1 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  -gcsg 17785  Abelcabl 18554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-sbg 17788  df-cmn 18555  df-abl 18556 This theorem is referenced by:  ablnnncan1  18589  baerlem5alem2  37781
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