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Mirrors > Home > MPE Home > Th. List > grpasscan2 | Structured version Visualization version GIF version |
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
Ref | Expression |
---|---|
grplcan.b | ⊢ 𝐵 = (Base‘𝐺) |
grplcan.p | ⊢ + = (+g‘𝐺) |
grpasscan1.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpasscan2 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) | |
2 | simp2 1133 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | grplcan.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
4 | grpasscan1.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
5 | 3, 4 | grpinvcl 18151 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
6 | 5 | 3adant2 1127 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
7 | simp3 1134 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
8 | grplcan.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | 3, 8 | grpass 18112 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (𝑋 + ((𝑁‘𝑌) + 𝑌))) |
10 | 1, 2, 6, 7, 9 | syl13anc 1368 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (𝑋 + ((𝑁‘𝑌) + 𝑌))) |
11 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 3, 8, 11, 4 | grplinv 18152 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺)) |
13 | 12 | 3adant2 1127 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺)) |
14 | 13 | oveq2d 7172 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑌) + 𝑌)) = (𝑋 + (0g‘𝐺))) |
15 | 3, 8, 11 | grprid 18134 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋) |
16 | 15 | 3adant3 1128 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋) |
17 | 10, 14, 16 | 3eqtrd 2860 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 invgcminusg 18104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 |
This theorem is referenced by: mulgaddcomlem 18250 |
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