![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18853 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) |
4 | 3 | fveq1d 6845 | . 2 ⊢ (𝐴 ∈ 𝑋 → ((𝐹‘𝐴)‘𝐵) = ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵)) |
5 | oveq2 7366 | . . 3 ⊢ (𝑎 = 𝐵 → (𝐴 + 𝑎) = (𝐴 + 𝐵)) | |
6 | eqid 2733 | . . 3 ⊢ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) | |
7 | ovex 7391 | . . 3 ⊢ (𝐴 + 𝐵) ∈ V | |
8 | 5, 6, 7 | fvmpt 6949 | . 2 ⊢ (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))‘𝐵) = (𝐴 + 𝐵)) |
9 | 4, 8 | sylan9eq 2793 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴)‘𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 |
This theorem is referenced by: cayleylem2 19200 dchrsum2 26632 sumdchr2 26634 |
Copyright terms: Public domain | W3C validator |