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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 20070 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 4003 | . . 3 ⊢ Ring ⊆ V | |
3 | fnssres 6673 | . . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring) | |
4 | 1, 2, 3 | mp2an 691 | . 2 ⊢ (mulGrp ↾ Ring) Fn Ring |
5 | fvres 6911 | . . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2728 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | ringmgp 20173 | . . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd) |
8 | 5, 7 | eqeltrd 2829 | . . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd) |
9 | 8 | rgen 3059 | . 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd |
10 | ffnfv 7124 | . 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 2099 ∀wral 3057 Vcvv 3470 ⊆ wss 3945 ↾ cres 5675 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 Mndcmnd 18688 mulGrpcmgp 20068 Ringcrg 20167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-mgp 20069 df-ring 20169 |
This theorem is referenced by: prdsringd 20251 prds1 20253 |
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