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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoismgmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndmgm 18704 as of 3-Feb-2020. A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mndoismgmOLD | ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndoissmgrpOLD 37472 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) | |
2 | smgrpismgmOLD 37466 | . 2 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Magmacmagm 37452 SemiGrpcsem 37464 MndOpcmndo 37470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-in 3951 df-sgrOLD 37465 df-mndo 37471 |
This theorem is referenced by: mndomgmid 37475 rngo1cl 37543 isdrngo2 37562 |
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