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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 20200 analog). (Contributed by AV, 22-Feb-2025.) |
Ref | Expression |
---|---|
rngmgpf | ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmgp 20088 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 4001 | . . 3 ⊢ Rng ⊆ V | |
3 | fnssres 6679 | . . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (mulGrp ↾ Rng) Fn Rng |
5 | fvres 6915 | . . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2725 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | rngmgp 20108 | . . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp) |
8 | 5, 7 | eqeltrd 2825 | . . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp) |
9 | 8 | rgen 3052 | . 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp |
10 | ffnfv 7128 | . 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 3050 Vcvv 3461 ⊆ wss 3944 ↾ cres 5680 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 Smgrpcsgrp 18681 mulGrpcmgp 20086 Rngcrng 20104 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-mgp 20087 df-rng 20105 |
This theorem is referenced by: prdsrngd 20128 |
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