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Mirrors > Home > MPE Home > Th. List > grpasscan1 | Structured version Visualization version GIF version |
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.) |
Ref | Expression |
---|---|
grplcan.b | ⊢ 𝐵 = (Base‘𝐺) |
grplcan.p | ⊢ + = (+g‘𝐺) |
grpasscan1.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpasscan1 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplcan.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grplcan.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | eqid 2725 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | grpasscan1.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
5 | 1, 2, 3, 4 | grprinv 18951 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺)) |
6 | 5 | 3adant3 1129 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺)) |
7 | 6 | oveq1d 7431 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = ((0g‘𝐺) + 𝑌)) |
8 | 1, 4 | grpinvcl 18948 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
9 | 1, 2 | grpass 18903 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))) |
10 | 9 | 3exp2 1351 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))))) |
11 | 10 | imp 405 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))) |
12 | 8, 11 | mpd 15 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))) |
13 | 12 | 3impia 1114 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))) |
14 | 1, 2, 3 | grplid 18928 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌) |
15 | 14 | 3adant2 1128 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌) |
16 | 7, 13, 15 | 3eqtr3d 2773 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 0gc0g 17420 Grpcgrp 18894 invgcminusg 18895 |
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-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-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-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 |
This theorem is referenced by: mulgaddcomlem 19056 ghmquskerlem1 19238 grplsmid 33162 nsgqusf1olem3 33174 ghmqusnsglem1 33178 qsdrnglem2 33256 |
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