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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 39675 | . . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} | |
2 | 1 | ssrab3 4059 | . 2 ⊢ LFinGen ⊆ LMod |
3 | 2 | sseli 3965 | 1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3937 𝒫 cpw 4541 “ cima 5560 ‘cfv 6357 Fincfn 8511 Basecbs 16485 LModclmod 19636 LSpanclspn 19745 LFinGenclfig 39674 |
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 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-in 3945 df-ss 3954 df-lfig 39675 |
This theorem is referenced by: lnrfg 39726 |
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