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Theorem ablpnpcan 18432
Description: Cancellation law for mixed addition and subtraction. (pnpcan 10522 analog.) (Contributed by NM, 29-May-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)
ablpnpcan.g (𝜑𝐺 ∈ Abel)
ablpnpcan.x (𝜑𝑋𝐵)
ablpnpcan.y (𝜑𝑌𝐵)
ablpnpcan.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ablpnpcan (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))

Proof of Theorem ablpnpcan
StepHypRef Expression
1 ablsubsub.g . . 3 (𝜑𝐺 ∈ Abel)
2 ablsubsub.x . . 3 (𝜑𝑋𝐵)
3 ablsubsub.y . . 3 (𝜑𝑌𝐵)
4 ablsubsub.z . . 3 (𝜑𝑍𝐵)
5 ablsubadd.b . . . 4 𝐵 = (Base‘𝐺)
6 ablsubadd.p . . . 4 + = (+g𝐺)
7 ablsubadd.m . . . 4 = (-g𝐺)
85, 6, 7ablsub4 18425 . . 3 ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋𝐵𝑍𝐵)) → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
91, 2, 3, 2, 4, 8syl122anc 1485 . 2 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = ((𝑋 𝑋) + (𝑌 𝑍)))
10 ablgrp 18405 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
111, 10syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
12 eqid 2771 . . . . 5 (0g𝐺) = (0g𝐺)
135, 12, 7grpsubid 17707 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 𝑋) = (0g𝐺))
1411, 2, 13syl2anc 573 . . 3 (𝜑 → (𝑋 𝑋) = (0g𝐺))
1514oveq1d 6808 . 2 (𝜑 → ((𝑋 𝑋) + (𝑌 𝑍)) = ((0g𝐺) + (𝑌 𝑍)))
165, 7grpsubcl 17703 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1711, 3, 4, 16syl3anc 1476 . . 3 (𝜑 → (𝑌 𝑍) ∈ 𝐵)
185, 6, 12grplid 17660 . . 3 ((𝐺 ∈ Grp ∧ (𝑌 𝑍) ∈ 𝐵) → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
1911, 17, 18syl2anc 573 . 2 (𝜑 → ((0g𝐺) + (𝑌 𝑍)) = (𝑌 𝑍))
209, 15, 193eqtrd 2809 1 (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630  -gcsg 17632  Abelcabl 18401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-cmn 18402  df-abl 18403
This theorem is referenced by:  hdmaprnlem7N  37665
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