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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 11529 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 19769 | . . 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 1376 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍))) |
10 | ablgrp 19744 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 1, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | eqid 2725 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
13 | 5, 12, 7 | grpsubid 18984 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
14 | 11, 2, 13 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
15 | 14 | oveq1d 7431 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍))) |
16 | 5, 7 | grpsubcl 18980 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
17 | 11, 3, 4, 16 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵) |
18 | 5, 6, 12 | grplid 18928 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
19 | 11, 17, 18 | syl2anc 582 | . 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
20 | 9, 15, 19 | 3eqtrd 2769 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 0gc0g 17420 Grpcgrp 18894 -gcsg 18896 Abelcabl 19740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 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 5144 df-opab 5206 df-mpt 5227 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 |
This theorem is referenced by: hdmaprnlem7N 41384 |
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