|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > mgpf | Structured version Visualization version GIF version | ||
| Description: Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| mgpf | ⊢ (mulGrp ↾ Ring):Ring⟶Mnd | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnmgp 20139 | . . 3 ⊢ mulGrp Fn V | |
| 2 | ssv 4008 | . . 3 ⊢ Ring ⊆ V | |
| 3 | fnssres 6691 | . . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring) | |
| 4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ (mulGrp ↾ Ring) Fn Ring | 
| 5 | fvres 6925 | . . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎)) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
| 7 | 6 | ringmgp 20236 | . . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd) | 
| 8 | 5, 7 | eqeltrd 2841 | . . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd) | 
| 9 | 8 | rgen 3063 | . 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd | 
| 10 | ffnfv 7139 | . 2 ⊢ ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)) | |
| 11 | 4, 9, 10 | mpbir2an 711 | 1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Mndcmnd 18747 mulGrpcmgp 20137 Ringcrg 20230 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-mgp 20138 df-ring 20232 | 
| This theorem is referenced by: prdsringd 20318 prds1 20320 | 
| Copyright terms: Public domain | W3C validator |