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Mirrors > Home > MPE Home > Th. List > grplactf1o | Structured version Visualization version GIF version |
Description: The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
grplact.1 | ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
grplact.2 | ⊢ 𝑋 = (Base‘𝐺) |
grplact.3 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
grplactf1o | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplact.1 | . . 3 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
2 | grplact.2 | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | grplact.3 | . . 3 ⊢ + = (+g‘𝐺) | |
4 | eqid 2760 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
5 | 1, 2, 3, 4 | grplactcnv 17719 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘((invg‘𝐺)‘𝐴)))) |
6 | 5 | simpld 477 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴):𝑋–1-1-onto→𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ↦ cmpt 4881 ◡ccnv 5265 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 +gcplusg 16143 Grpcgrp 17623 invgcminusg 17624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-0g 16304 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 |
This theorem is referenced by: eqgen 17848 dchrsum2 25192 sumdchr2 25194 |
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