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Theorem ablsubsub4 19738
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 19705 . . . . 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 18948 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
93, 4, 5, 8syl3anc 1368 . . 3 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
10 ablsubsub.z . . 3 (𝜑𝑍𝐵)
11 ablsubadd.p . . . 4 + = (+g𝐺)
12 eqid 2726 . . . 4 (invg𝐺) = (invg𝐺)
136, 11, 12, 7grpsubval 18915 . . 3 (((𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
149, 10, 13syl2anc 583 . 2 (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
156, 12grpinvcl 18917 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
163, 10, 15syl2anc 583 . . 3 (𝜑 → ((invg𝐺)‘𝑍) ∈ 𝐵)
176, 11, 7, 1, 4, 5, 16ablsubsub 19737 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = ((𝑋 𝑌) + ((invg𝐺)‘𝑍)))
186, 11, 7, 12, 3, 5, 10grpsubinv 18941 . . 3 (𝜑 → (𝑌 ((invg𝐺)‘𝑍)) = (𝑌 + 𝑍))
1918oveq2d 7421 . 2 (𝜑 → (𝑋 (𝑌 ((invg𝐺)‘𝑍))) = (𝑋 (𝑌 + 𝑍)))
2014, 17, 193eqtr2d 2772 1 (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863  invgcminusg 18864  -gcsg 18865  Abelcabl 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-cmn 19702  df-abl 19703
This theorem is referenced by:  ablsub32  19741  ablnnncan  19742  rngqiprngfulem4  21167  ip2subdi  21537  cpmadugsumlemF  22733  baerlem5alem2  41095
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