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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 37857 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) | |
2 | mndoisexid 37856 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) | |
3 | 1, 2 | elind 4210 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3962 ExId cexid 37831 Magmacmagm 37835 MndOpcmndo 37853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-in 3970 df-sgrOLD 37848 df-mndo 37854 |
This theorem is referenced by: ismndo2 37861 rngoidmlem 37923 isdrngo2 37945 |
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