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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 18314 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 3588 | . . 3 ⊢ Ring ⊆ V | |
3 | fnssres 5918 | . . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring) | |
4 | 1, 2, 3 | mp2an 704 | . 2 ⊢ (mulGrp ↾ Ring) Fn Ring |
5 | fvres 6117 | . . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2610 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | ringmgp 18376 | . . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd) |
8 | 5, 7 | eqeltrd 2688 | . . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd) |
9 | 8 | rgen 2906 | . 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd |
10 | ffnfv 6295 | . 2 ⊢ ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)) | |
11 | 4, 9, 10 | mpbir2an 957 | 1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Mndcmnd 17117 mulGrpcmgp 18312 Ringcrg 18370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-mgp 18313 df-ring 18372 |
This theorem is referenced by: prdsringd 18435 prds1 18437 |
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