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Mirrors > Home > MPE Home > Th. List > ablnncan | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nncan 10652 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) |
Ref | Expression |
---|---|
ablnncan | ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2778 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ablnncan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ablnncan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 5, 6 | ablsubsub 18609 | . 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌)) |
8 | ablgrp 18584 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | eqid 2778 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
11 | 1, 10, 3 | grpsubid 17886 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
12 | 9, 5, 11 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
13 | 12 | oveq1d 6937 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌)) |
14 | 1, 2, 10 | grplid 17839 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
15 | 9, 6, 14 | syl2anc 579 | . 2 ⊢ (𝜑 → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
16 | 7, 13, 15 | 3eqtrd 2818 | 1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Grpcgrp 17809 -gcsg 17811 Abelcabl 18580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-cmn 18581 df-abl 18582 |
This theorem is referenced by: ablnnncan1 18615 pgpfac1lem3 18863 tsmsxplem1 22364 baerlem5blem2 37868 |
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