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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 20266 analog). (Contributed by AV, 22-Feb-2025.) |
Ref | Expression |
---|---|
rngmgpf | ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmgp 20154 | . . 3 ⊢ mulGrp Fn V | |
2 | ssv 4020 | . . 3 ⊢ Rng ⊆ V | |
3 | fnssres 6692 | . . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ (mulGrp ↾ Rng) Fn Rng |
5 | fvres 6926 | . . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎)) | |
6 | eqid 2735 | . . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎) | |
7 | 6 | rngmgp 20174 | . . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp) |
8 | 5, 7 | eqeltrd 2839 | . . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp) |
9 | 8 | rgen 3061 | . 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp |
10 | ffnfv 7139 | . 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 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 ↾ cres 5691 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Smgrpcsgrp 18744 mulGrpcmgp 20152 Rngcrng 20170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-mgp 20153 df-rng 20171 |
This theorem is referenced by: prdsrngd 20194 |
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