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Mirrors > Home > MPE Home > Th. List > Mathboxes > smgrpismgmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sgrpmgm 18749 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 3978 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass)) | |
2 | 1 | simplbi 497 | . 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma) |
3 | df-sgrOLD 37847 | . 2 ⊢ SemiGrp = (Magma ∩ Ass) | |
4 | 2, 3 | eleq2s 2856 | 1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3961 Asscass 37828 Magmacmagm 37834 SemiGrpcsem 37846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-in 3969 df-sgrOLD 37847 |
This theorem is referenced by: mndoismgmOLD 37856 |
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