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Mirrors > Home > MPE Home > Th. List > ablnnncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nnncan1 10911 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 18903 | . . . . 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 18171 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) ∈ 𝐵) |
10 | 7, 4, 8, 9 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 10 | ablsub32 18935 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = ((𝑋 − (𝑋 − 𝑍)) − 𝑌)) |
12 | 1, 2, 3, 4, 8 | ablnncan 18934 | . . 3 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑍)) = 𝑍) |
13 | 12 | oveq1d 7150 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑋 − 𝑍)) − 𝑌) = (𝑍 − 𝑌)) |
14 | 11, 13 | eqtrd 2833 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Grpcgrp 18095 -gcsg 18097 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: minveclem2 24030 ply1divmo 24736 baerlem3lem2 39006 |
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