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| Mirrors > Home > MPE Home > Th. List > Mathboxes > smgrpismgmOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of sgrpmgm 18737 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| smgrpismgmOLD | ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3967 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass)) | |
| 2 | 1 | simplbi 497 | . 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma) |
| 3 | df-sgrOLD 37868 | . 2 ⊢ SemiGrp = (Magma ∩ Ass) | |
| 4 | 2, 3 | eleq2s 2859 | 1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 Asscass 37849 Magmacmagm 37855 SemiGrpcsem 37867 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-sgrOLD 37868 |
| This theorem is referenced by: mndoismgmOLD 37877 |
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