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Mirrors > Home > MPE Home > Th. List > rngmgpf | Structured version Visualization version GIF version |
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20195 analog). (Contributed by AV, 22-Feb-2025.) |
Ref | Expression |
---|---|
rngmgpf | ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmgp 20083 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 4006 | . . 3 ⊢ Rng ⊆ V | |
3 | fnssres 6683 | . . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (mulGrp ↾ Rng) Fn Rng |
5 | fvres 6921 | . . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2728 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | rngmgp 20103 | . . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp) |
8 | 5, 7 | eqeltrd 2829 | . . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp) |
9 | 8 | rgen 3060 | . 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp |
10 | ffnfv 7134 | . 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)) | |
11 | 4, 9, 10 | mpbir2an 709 | 1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ⊆ wss 3949 ↾ cres 5684 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 Smgrpcsgrp 18685 mulGrpcmgp 20081 Rngcrng 20099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-mgp 20082 df-rng 20100 |
This theorem is referenced by: prdsrngd 20123 |
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