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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 35147 | . . 3 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐺 ∈ MndOp ↔ 𝐺 ∈ (SemiGrp ∩ ExId )) |
3 | elin 4171 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
4 | ismndo.1 | . . . . 5 ⊢ 𝑋 = dom dom 𝐺 | |
5 | 4 | isexid 35127 | . . . 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 | syl5bb 285 | . 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
8 | 2, 7 | syl5bb 285 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 dom cdm 5557 (class class class)co 7158 ExId cexid 35124 SemiGrpcsem 35140 MndOpcmndo 35146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 df-exid 35125 df-mndo 35147 |
This theorem is referenced by: ismndo1 35153 |
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