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Mirrors > Home > MPE Home > Th. List > grplactfval | Structured version Visualization version GIF version |
Description: The left group action of element 𝐴 of group 𝐺. (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 |
---|---|
grplactfval | ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7165 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑔 + 𝑎) = (𝐴 + 𝑎)) | |
2 | 1 | mpteq2dv 5164 | . 2 ⊢ (𝑔 = 𝐴 → (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) |
3 | grplact.1 | . 2 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
4 | grplact.2 | . 2 ⊢ 𝑋 = (Base‘𝐺) | |
5 | 2, 3, 4 | mptfvmpt 6992 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 |
This theorem is referenced by: grplactval 18203 grplactcnv 18204 eqglact 18333 eqgen 18335 tgplacthmeo 22713 tgpconncompeqg 22722 |
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