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Theorem ablsub32 18933
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 2822 . . . . 5 (+g𝐺) = (+g𝐺)
64, 5ablcom 18915 . . . 4 ((𝐺 ∈ Abel ∧ 𝑌𝐵𝑍𝐵) → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
71, 2, 3, 6syl3anc 1368 . . 3 (𝜑 → (𝑌(+g𝐺)𝑍) = (𝑍(+g𝐺)𝑌))
87oveq2d 7156 . 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 18930 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌(+g𝐺)𝑍)))
124, 5, 9, 1, 10, 3, 2ablsubsub4 18930 . 2 (𝜑 → ((𝑋 𝑍) 𝑌) = (𝑋 (𝑍(+g𝐺)𝑌)))
138, 11, 123eqtr4d 2867 1 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  -gcsg 18096  Abelcabl 18898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-sbg 18099  df-cmn 18899  df-abl 18900
This theorem is referenced by:  ablnnncan1  18935  baerlem5alem2  38965
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