![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > smgrpismgmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sgrpmgm 17675 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 4019 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass)) | |
2 | 1 | simplbi 493 | . 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma) |
3 | df-sgrOLD 34286 | . 2 ⊢ SemiGrp = (Magma ∩ Ass) | |
4 | 2, 3 | eleq2s 2877 | 1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3791 Asscass 34267 Magmacmagm 34273 SemiGrpcsem 34285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-v 3400 df-in 3799 df-sgrOLD 34286 |
This theorem is referenced by: mndoismgmOLD 34295 |
Copyright terms: Public domain | W3C validator |