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Mirrors > Home > MPE Home > Th. List > Mathboxes > smgrpismgmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sgrpmgm 17112 as of 3-Feb-2020. A semi-group 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 3758 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass)) | |
2 | 1 | simplbi 475 | . 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma) |
3 | df-sgrOLD 32830 | . 2 ⊢ SemiGrp = (Magma ∩ Ass) | |
4 | 2, 3 | eleq2s 2706 | 1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∩ cin 3539 Asscass 32811 Magmacmagm 32817 SemiGrpcsem 32829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-sgrOLD 32830 |
This theorem is referenced by: mndoismgmOLD 32839 |
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