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Theorem abladdsub 19732
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
abladdsub ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))

Proof of Theorem abladdsub
StepHypRef Expression
1 ablsubadd.b . . . . 5 𝐵 = (Base‘𝐺)
2 ablsubadd.p . . . . 5 + = (+g𝐺)
31, 2ablcom 19719 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
433adant3r3 1181 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
54oveq1d 7420 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑌 + 𝑋) 𝑍))
6 ablgrp 19705 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
76adantr 480 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
8 simpr2 1192 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
9 simpr1 1191 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
10 simpr3 1193 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
11 ablsubadd.m . . . 4 = (-g𝐺)
121, 2, 11grpaddsubass 18958 . . 3 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑋𝐵𝑍𝐵)) → ((𝑌 + 𝑋) 𝑍) = (𝑌 + (𝑋 𝑍)))
137, 8, 9, 10, 12syl13anc 1369 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 + 𝑋) 𝑍) = (𝑌 + (𝑋 𝑍)))
14 simpl 482 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Abel)
151, 11grpsubcl 18948 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
167, 9, 10, 15syl3anc 1368 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
171, 2ablcom 19719 . . 3 ((𝐺 ∈ Abel ∧ 𝑌𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → (𝑌 + (𝑋 𝑍)) = ((𝑋 𝑍) + 𝑌))
1814, 8, 16, 17syl3anc 1368 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 + (𝑋 𝑍)) = ((𝑋 𝑍) + 𝑌))
195, 13, 183eqtrd 2770 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863  -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:  ablsubadd23  19733  ablpncan2  19735  ablsubsub  19737  ip2subdi  21537  r1padd1  33183
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