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Theorem ablsubaddsub 19847
Description: Double subtraction and addition in abelian groups. (cnambpcma 47244 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 19846 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑍) = (𝑋 + (𝑍 𝑌)))
54oveq1d 7446 . 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 19818 . . . . . 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 19051 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
139, 10, 11, 12syl3anc 1370 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑌) ∈ 𝐵)
141, 2ablcom 19832 . . . 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 7446 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + (𝑍 𝑌)) 𝑋) = (((𝑍 𝑌) + 𝑋) 𝑋))
171, 2, 3grpaddsubass 19061 . . . 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 2735 . . . . . 6 (0g𝐺) = (0g𝐺)
201, 19, 3grpsubid 19055 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
219, 7, 20syl2anc 584 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋) = (0g𝐺))
2221oveq2d 7447 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (𝑋 𝑋)) = ((𝑍 𝑌) + (0g𝐺)))
231, 2, 19, 9, 13grpridd 19001 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (0g𝐺)) = (𝑍 𝑌))
2418, 22, 233eqtrd 2779 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = (𝑍 𝑌))
255, 16, 243eqtrd 2779 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 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  Abelcabl 19814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-cmn 19815  df-abl 19816
This theorem is referenced by:  rngqiprngimfo  21329
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