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Theorem ablsubaddsub 19724
Description: Double subtraction and addition in abelian groups. (cnambpcma 46301 analog.) (Contributed by AV, 3-Mar-2025.)
Hypotheses
Ref Expression
ablsubadd.b 𝐵 = (Base‘𝐺)
ablsubadd.p + = (+g𝐺)
ablsubadd.m = (-g𝐺)
Assertion
Ref Expression
ablsubaddsub ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = (𝑍 𝑌))

Proof of Theorem ablsubaddsub
StepHypRef Expression
1 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
2 ablsubadd.p . . . 4 + = (+g𝐺)
3 ablsubadd.m . . . 4 = (-g𝐺)
41, 2, 3ablsubadd23 19723 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑍) = (𝑋 + (𝑍 𝑌)))
54oveq1d 7427 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = ((𝑋 + (𝑍 𝑌)) 𝑋))
6 simpl 482 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Abel)
7 simpr1 1193 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
8 ablgrp 19695 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
98adantr 480 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr3 1195 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
11 simpr2 1194 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
121, 3grpsubcl 18940 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
139, 10, 11, 12syl3anc 1370 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑌) ∈ 𝐵)
141, 2ablcom 19709 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
156, 7, 13, 14syl3anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
1615oveq1d 7427 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + (𝑍 𝑌)) 𝑋) = (((𝑍 𝑌) + 𝑋) 𝑋))
171, 2, 3grpaddsubass 18950 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑍 𝑌) ∈ 𝐵𝑋𝐵𝑋𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
189, 13, 7, 7, 17syl13anc 1371 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
19 eqid 2731 . . . . . 6 (0g𝐺) = (0g𝐺)
201, 19, 3grpsubid 18944 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
219, 7, 20syl2anc 583 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋) = (0g𝐺))
2221oveq2d 7428 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (𝑋 𝑋)) = ((𝑍 𝑌) + (0g𝐺)))
231, 2, 19, 9, 13grpridd 18892 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (0g𝐺)) = (𝑍 𝑌))
2418, 22, 233eqtrd 2775 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = (𝑍 𝑌))
255, 16, 243eqtrd 2775 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  0gc0g 17390  Grpcgrp 18856  -gcsg 18858  Abelcabl 19691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-sbg 18861  df-cmn 19692  df-abl 19693
This theorem is referenced by:  rngqiprngimfo  21061
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