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Mirrors > Home > MPE Home > Th. List > grplactval | Structured version Visualization version GIF version |
Description: The value of the left group action of element 𝐴 of group 𝐺 at 𝐵. (Contributed by Paul Chapman, 18-Mar-2008.) |
Ref | Expression |
---|---|
grplact.1 | ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
grplact.2 | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
grplactval | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplact.1 | . . . 4 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
2 | grplact.2 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
3 | 1, 2 | grplactfval 17869 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) |
4 | 3 | fveq1d 6434 | . 2 ⊢ (𝐴 ∈ 𝑋 → ((𝐹‘𝐴)‘𝐵) = ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵)) |
5 | oveq2 6912 | . . 3 ⊢ (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵)) | |
6 | eqid 2824 | . . 3 ⊢ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) | |
7 | ovex 6936 | . . 3 ⊢ (𝐴 + 𝐵) ∈ V | |
8 | 5, 6, 7 | fvmpt 6528 | . 2 ⊢ (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵)) |
9 | 4, 8 | sylan9eq 2880 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ↦ cmpt 4951 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 |
This theorem is referenced by: cayleylem2 18182 dchrsum2 25405 sumdchr2 25407 |
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