| Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ismndo | Structured version Visualization version GIF version | ||
| Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ismndo.1 | ⊢ 𝑋 = dom dom 𝐺 |
| Ref | Expression |
|---|---|
| ismndo | ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-mndo 37874 | . . 3 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐺 ∈ MndOp ↔ 𝐺 ∈ (SemiGrp ∩ ExId )) |
| 3 | elin 3967 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
| 4 | ismndo.1 | . . . . 5 ⊢ 𝑋 = dom dom 𝐺 | |
| 5 | 4 | isexid 37854 | . . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
| 6 | 5 | anbi2d 630 | . . 3 ⊢ (𝐺 ∈ 𝐴 → ((𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
| 7 | 3, 6 | bitrid 283 | . 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
| 8 | 2, 7 | bitrid 283 | 1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 dom cdm 5685 (class class class)co 7431 ExId cexid 37851 SemiGrpcsem 37867 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-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-exid 37852 df-mndo 37874 |
| This theorem is referenced by: ismndo1 37880 |
| Copyright terms: Public domain | W3C validator |