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Mirrors > Home > MPE Home > Th. List > Mathboxes > fglmod | Structured version Visualization version GIF version |
Description: Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
fglmod | ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lfig 39688 | . . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} | |
2 | 1 | ssrab3 4057 | . 2 ⊢ LFinGen ⊆ LMod |
3 | 2 | sseli 3963 | 1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3935 𝒫 cpw 4539 “ cima 5558 ‘cfv 6355 Fincfn 8509 Basecbs 16483 LModclmod 19634 LSpanclspn 19743 LFinGenclfig 39687 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-in 3943 df-ss 3952 df-lfig 39688 |
This theorem is referenced by: lnrfg 39739 |
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