|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ablnnncan1 | Structured version Visualization version GIF version | ||
| Description: Cancellation law for group subtraction. (nnncan1 11546 analog.) (Contributed by NM, 7-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| ablnncan.b | ⊢ 𝐵 = (Base‘𝐺) | 
| ablnncan.m | ⊢ − = (-g‘𝐺) | 
| ablnncan.g | ⊢ (𝜑 → 𝐺 ∈ Abel) | 
| ablnncan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| ablnncan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| ablsub32.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| ablnnncan1 | ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
| 3 | ablnncan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
| 4 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | ablnncan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | ablgrp 19804 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 8 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 9 | 1, 2 | grpsubcl 19039 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) ∈ 𝐵) | 
| 10 | 7, 4, 8, 9 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝐵) | 
| 11 | 1, 2, 3, 4, 5, 10 | ablsub32 19840 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = ((𝑋 − (𝑋 − 𝑍)) − 𝑌)) | 
| 12 | 1, 2, 3, 4, 8 | ablnncan 19839 | . . 3 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑍)) = 𝑍) | 
| 13 | 12 | oveq1d 7447 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑋 − 𝑍)) − 𝑌) = (𝑍 − 𝑌)) | 
| 14 | 11, 13 | eqtrd 2776 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Grpcgrp 18952 -gcsg 18954 Abelcabl 19800 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-cmn 19801 df-abl 19802 | 
| This theorem is referenced by: minveclem2 25461 ply1divmo 26176 baerlem3lem2 41713 | 
| Copyright terms: Public domain | W3C validator |