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Theorem ablsubsub4 18543
Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
ablsubsub.g (𝜑𝐺 ∈ Abel)
ablsubsub.x (𝜑𝑋𝐵)
ablsubsub.y (𝜑𝑌𝐵)
ablsubsub.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablsubsub4 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))

Proof of Theorem ablsubsub4
StepHypRef Expression
1 ablsubsub.g . . . . 5 (𝜑𝐺 ∈ Abel)
2 ablgrp 18517 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
31, 2syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
4 ablsubsub.x . . . 4 (𝜑𝑋𝐵)
5 ablsubsub.y . . . 4 (𝜑𝑌𝐵)
6 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
7 ablsubadd.m . . . . 5 = (-g𝐺)
86, 7grpsubcl 17815 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1491 . . 3 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
10 ablsubsub.z . . 3 (𝜑𝑍𝐵)
11 ablsubadd.p . . . 4 + = (+g𝐺)
12 eqid 2803 . . . 4 (invg𝐺) = (invg𝐺)
136, 11, 12, 7grpsubval 17785 . . 3 (((𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
149, 10, 13syl2anc 580 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
156, 12grpinvcl 17787 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
163, 10, 15syl2anc 580 . . 3 (𝜑 → ((invg𝐺)‘𝑍) ∈ 𝐵)
176, 11, 7, 1, 4, 5, 16ablsubsub 18542 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
186, 11, 7, 12, 3, 5, 10grpsubinv 17808 . . 3 (𝜑 → (𝑌 ((invg𝐺)‘𝑍)) = (𝑌 + 𝑍))
1918oveq2d 6898 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = (𝑋 (𝑌 + 𝑍)))
2014, 17, 193eqtr2d 2843 1 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cfv 6105  (class class class)co 6882  Basecbs 16188  +gcplusg 16271  Grpcgrp 17742  invgcminusg 17743  -gcsg 17744  Abelcabl 18513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-riota 6843  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-1st 7405  df-2nd 7406  df-0g 16421  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-minusg 17746  df-sbg 17747  df-cmn 18514  df-abl 18515
This theorem is referenced by:  ablsub32  18546  ablnnncan  18547  ip2subdi  20317  cpmadugsumlemF  21013  baerlem5alem2  37736
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