![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rnrhmsubrg | Structured version Visualization version GIF version |
Description: The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.) |
Ref | Expression |
---|---|
rnrhmsubrg | ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑁) = (Base‘𝑁) | |
3 | 1, 2 | rhmf 20501 | . . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁)) |
4 | 3 | ffnd 6737 | . . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 Fn (Base‘𝑀)) |
5 | fnima 6698 | . . 3 ⊢ (𝐹 Fn (Base‘𝑀) → (𝐹 “ (Base‘𝑀)) = ran 𝐹) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ (Base‘𝑀)) = ran 𝐹) |
7 | rhmrcl1 20492 | . . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑀 ∈ Ring) | |
8 | 1 | subrgid 20589 | . . . 4 ⊢ (𝑀 ∈ Ring → (Base‘𝑀) ∈ (SubRing‘𝑀)) |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (Base‘𝑀) ∈ (SubRing‘𝑀)) |
10 | rhmima 20620 | . . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ (Base‘𝑀) ∈ (SubRing‘𝑀)) → (𝐹 “ (Base‘𝑀)) ∈ (SubRing‘𝑁)) | |
11 | 9, 10 | mpdan 687 | . 2 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ (Base‘𝑀)) ∈ (SubRing‘𝑁)) |
12 | 6, 11 | eqeltrrd 2839 | 1 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ran crn 5689 “ cima 5691 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Ringcrg 20250 RingHom crh 20485 SubRingcsubrg 20585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-subg 19153 df-ghm 19243 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-rhm 20488 df-subrng 20562 df-subrg 20586 |
This theorem is referenced by: rnasclsubrg 21930 idomsubr 33290 algextdeglem4 33725 selvcllem3 42565 |
Copyright terms: Public domain | W3C validator |