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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 19906 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 3972 | . . 3 ⊢ Ring ⊆ V | |
3 | fnssres 6628 | . . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring) | |
4 | 1, 2, 3 | mp2an 691 | . 2 ⊢ (mulGrp ↾ Ring) Fn Ring |
5 | fvres 6865 | . . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2733 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | ringmgp 19978 | . . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd) |
8 | 5, 7 | eqeltrd 2834 | . . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd) |
9 | 8 | rgen 3063 | . 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd |
10 | ffnfv 7070 | . 2 ⊢ ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)) | |
11 | 4, 9, 10 | mpbir2an 710 | 1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∀wral 3061 Vcvv 3447 ⊆ wss 3914 ↾ cres 5639 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 Mndcmnd 18564 mulGrpcmgp 19904 Ringcrg 19972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-mgp 19905 df-ring 19974 |
This theorem is referenced by: prdsringd 20044 prds1 20046 |
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