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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mndomgmid | Structured version Visualization version GIF version | ||
| Description: A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| mndomgmid | ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndoismgmOLD 37877 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) | |
| 2 | mndoisexid 37876 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) | |
| 3 | 1, 2 | elind 4200 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 ExId cexid 37851 Magmacmagm 37855 MndOpcmndo 37873 |
| 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 df-mndo 37874 |
| This theorem is referenced by: ismndo2 37881 rngoidmlem 37943 isdrngo2 37965 |
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