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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 43080 | . . 3 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} | |
| 2 | 1 | ssrab3 4082 | . 2 ⊢ LFinGen ⊆ LMod |
| 3 | 2 | sseli 3979 | 1 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 𝒫 cpw 4600 “ cima 5688 ‘cfv 6561 Fincfn 8985 Basecbs 17247 LModclmod 20858 LSpanclspn 20969 LFinGenclfig 43079 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-ss 3968 df-lfig 43080 |
| This theorem is referenced by: lnrfg 43131 |
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