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Theorem ablsubaddsub 19753
Description: Double subtraction and addition in abelian groups. (cnambpcma 46587 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 19752 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) + 𝑍) = (𝑋 + (𝑍 𝑌)))
54oveq1d 7429 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = ((𝑋 + (𝑍 𝑌)) 𝑋))
6 simpl 482 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Abel)
7 simpr1 1192 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
8 ablgrp 19724 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
98adantr 480 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr3 1194 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
11 simpr2 1193 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
121, 3grpsubcl 18960 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵𝑌𝐵) → (𝑍 𝑌) ∈ 𝐵)
139, 10, 11, 12syl3anc 1369 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑌) ∈ 𝐵)
141, 2ablcom 19738 . . . 4 ((𝐺 ∈ Abel ∧ 𝑋𝐵 ∧ (𝑍 𝑌) ∈ 𝐵) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
156, 7, 13, 14syl3anc 1369 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑍 𝑌)) = ((𝑍 𝑌) + 𝑋))
1615oveq1d 7429 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + (𝑍 𝑌)) 𝑋) = (((𝑍 𝑌) + 𝑋) 𝑋))
171, 2, 3grpaddsubass 18970 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑍 𝑌) ∈ 𝐵𝑋𝐵𝑋𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
189, 13, 7, 7, 17syl13anc 1370 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = ((𝑍 𝑌) + (𝑋 𝑋)))
19 eqid 2727 . . . . . 6 (0g𝐺) = (0g𝐺)
201, 19, 3grpsubid 18964 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
219, 7, 20syl2anc 583 . . . 4 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋) = (0g𝐺))
2221oveq2d 7430 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (𝑋 𝑋)) = ((𝑍 𝑌) + (0g𝐺)))
231, 2, 19, 9, 13grpridd 18912 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑍 𝑌) + (0g𝐺)) = (𝑍 𝑌))
2418, 22, 233eqtrd 2771 . 2 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑍 𝑌) + 𝑋) 𝑋) = (𝑍 𝑌))
255, 16, 243eqtrd 2771 1 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌) + 𝑍) 𝑋) = (𝑍 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  Basecbs 17165  +gcplusg 17218  0gc0g 17406  Grpcgrp 18875  -gcsg 18877  Abelcabl 19720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-0g 17408  df-mgm 18585  df-sgrp 18664  df-mnd 18680  df-grp 18878  df-minusg 18879  df-sbg 18880  df-cmn 19721  df-abl 19722
This theorem is referenced by:  rngqiprngimfo  21173
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