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Mirrors > Home > MPE Home > Th. List > ablpnpcan | Structured version Visualization version GIF version |
Description: Cancellation law for mixed addition and subtraction. (pnpcan 10919 analog.) (Contributed by NM, 29-May-2015.) |
Ref | Expression |
---|---|
ablsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
ablsubadd.p | ⊢ + = (+g‘𝐺) |
ablsubadd.m | ⊢ − = (-g‘𝐺) |
ablsubsub.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablsubsub.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablsubsub.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ablsubsub.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ablpnpcan.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablpnpcan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablpnpcan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ablpnpcan.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ablpnpcan | ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsubsub.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablsubsub.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ablsubsub.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ablsubsub.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | ablsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
7 | ablsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 5, 6, 7 | ablsub4 18927 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍))) |
9 | 1, 2, 3, 2, 4, 8 | syl122anc 1375 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍))) |
10 | ablgrp 18905 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 1, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
13 | 5, 12, 7 | grpsubid 18177 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
14 | 11, 2, 13 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
15 | 14 | oveq1d 7165 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍))) |
16 | 5, 7 | grpsubcl 18173 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
17 | 11, 3, 4, 16 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵) |
18 | 5, 6, 12 | grplid 18127 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
19 | 11, 17, 18 | syl2anc 586 | . 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
20 | 9, 15, 19 | 3eqtrd 2860 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Grpcgrp 18097 -gcsg 18099 Abelcabl 18901 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: hdmaprnlem7N 38985 |
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