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Theorem ablsubsub4 18444
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 18418 . . . . 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 17716 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1477 . . 3 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
10 ablsubsub.z . . 3 (𝜑𝑍𝐵)
11 ablsubadd.p . . . 4 + = (+g𝐺)
12 eqid 2760 . . . 4 (invg𝐺) = (invg𝐺)
136, 11, 12, 7grpsubval 17686 . . 3 (((𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
149, 10, 13syl2anc 696 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
156, 12grpinvcl 17688 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
163, 10, 15syl2anc 696 . . 3 (𝜑 → ((invg𝐺)‘𝑍) ∈ 𝐵)
176, 11, 7, 1, 4, 5, 16ablsubsub 18443 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
186, 11, 7, 12, 3, 5, 10grpsubinv 17709 . . 3 (𝜑 → (𝑌 ((invg𝐺)‘𝑍)) = (𝑌 + 𝑍))
1918oveq2d 6830 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = (𝑋 (𝑌 + 𝑍)))
2014, 17, 193eqtr2d 2800 1 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  Grpcgrp 17643  invgcminusg 17644  -gcsg 17645  Abelcabl 18414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-sbg 17648  df-cmn 18415  df-abl 18416
This theorem is referenced by:  ablsub32  18447  ablnnncan  18448  ip2subdi  20211  cpmadugsumlemF  20903  baerlem5alem2  37520
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