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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 37241 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) | |
2 | mndoisexid 37240 | . 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) | |
3 | 1, 2 | elind 4187 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∩ cin 3940 ExId cexid 37215 Magmacmagm 37219 MndOpcmndo 37237 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3948 df-sgrOLD 37232 df-mndo 37238 |
This theorem is referenced by: ismndo2 37245 rngoidmlem 37307 isdrngo2 37329 |
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