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| Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20246 analog). (Contributed by AV, 22-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| rngmgpf | ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnmgp 20140 | . . 3 ⊢ mulGrp Fn V | |
| 2 | ssv 4007 | . . 3 ⊢ Rng ⊆ V | |
| 3 | fnssres 6690 | . . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng) | |
| 4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ (mulGrp ↾ Rng) Fn Rng | 
| 5 | fvres 6924 | . . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎)) | |
| 6 | eqid 2736 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
| 7 | 6 | rngmgp 20154 | . . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp) | 
| 8 | 5, 7 | eqeltrd 2840 | . . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp) | 
| 9 | 8 | rgen 3062 | . 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp | 
| 10 | ffnfv 7138 | . 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)) | |
| 11 | 4, 9, 10 | mpbir2an 711 | 1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 ↾ cres 5686 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 Smgrpcsgrp 18732 mulGrpcmgp 20138 Rngcrng 20150 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-mgp 20139 df-rng 20151 | 
| This theorem is referenced by: prdsrngd 20174 | 
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