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Mathbox for Jeff Madsen |
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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 37853 | . . 3 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐺 ∈ MndOp ↔ 𝐺 ∈ (SemiGrp ∩ ExId )) |
3 | elin 3978 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
4 | ismndo.1 | . . . . 5 ⊢ 𝑋 = dom dom 𝐺 | |
5 | 4 | isexid 37833 | . . . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ∩ cin 3961 dom cdm 5688 (class class class)co 7430 ExId cexid 37830 SemiGrpcsem 37846 MndOpcmndo 37852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-dm 5698 df-iota 6515 df-fv 6570 df-ov 7433 df-exid 37831 df-mndo 37853 |
This theorem is referenced by: ismndo1 37859 |
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