![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ablpnpcan | Structured version Visualization version GIF version |
Description: Cancellation law for mixed addition and subtraction. (pnpcan 11503 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 19730 | . . 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 19705 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 1, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | eqid 2726 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
13 | 5, 12, 7 | grpsubid 18952 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
14 | 11, 2, 13 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
15 | 14 | oveq1d 7420 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍))) |
16 | 5, 7 | grpsubcl 18948 | . . . 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 18897 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
19 | 11, 17, 18 | syl2anc 583 | . 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
20 | 9, 15, 19 | 3eqtrd 2770 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 0gc0g 17394 Grpcgrp 18863 -gcsg 18865 Abelcabl 19701 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 |
This theorem is referenced by: hdmaprnlem7N 41239 |
Copyright terms: Public domain | W3C validator |