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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 20150 analog). (Contributed by AV, 22-Feb-2025.) |
Ref | Expression |
---|---|
rngmgpf | ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmgp 20038 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 4001 | . . 3 ⊢ Rng ⊆ V | |
3 | fnssres 6666 | . . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng) | |
4 | 1, 2, 3 | mp2an 689 | . 2 ⊢ (mulGrp ↾ Rng) Fn Rng |
5 | fvres 6903 | . . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2726 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | rngmgp 20058 | . . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp) |
8 | 5, 7 | eqeltrd 2827 | . . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp) |
9 | 8 | rgen 3057 | . 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp |
10 | ffnfv 7113 | . 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)) | |
11 | 4, 9, 10 | mpbir2an 708 | 1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 ↾ cres 5671 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 Smgrpcsgrp 18648 mulGrpcmgp 20036 Rngcrng 20054 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-mgp 20037 df-rng 20055 |
This theorem is referenced by: prdsrngd 20078 |
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