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Mirrors > Home > MPE Home > Th. List > idrhm | Structured version Visualization version GIF version |
Description: The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.) |
Ref | Expression |
---|---|
idrhm.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
idrhm | ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
2 | ringgrp 20139 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | idrhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 3 | idghm 19152 | . . . 4 ⊢ (𝑅 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅)) |
6 | eqid 2724 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6 | ringmgp 20140 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
8 | 6, 3 | mgpbas 20041 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
9 | 8 | idmhm 18721 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → ( I ↾ 𝐵) ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑅))) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑅))) |
11 | 5, 10 | jca 511 | . 2 ⊢ (𝑅 ∈ Ring → (( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅) ∧ ( I ↾ 𝐵) ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑅)))) |
12 | 6, 6 | isrhm 20376 | . 2 ⊢ (( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ Ring) ∧ (( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅) ∧ ( I ↾ 𝐵) ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑅))))) |
13 | 1, 1, 11, 12 | syl21anbrc 1341 | 1 ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 I cid 5564 ↾ cres 5669 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 Mndcmnd 18663 MndHom cmhm 18707 Grpcgrp 18859 GrpHom cghm 19134 mulGrpcmgp 20035 Ringcrg 20134 RingHom crh 20367 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-grp 18862 df-ghm 19135 df-mgp 20036 df-ur 20083 df-ring 20136 df-rhm 20370 |
This theorem is referenced by: rhmsubcsetclem1 20552 rhmsubcrngclem1 20558 srhmsubc 20572 rhmsubclem3 20579 rhmsubcALTVlem3 47207 funcringcsetcALTV2lem7 47220 ringccatidALTV 47230 funcringcsetclem7ALTV 47243 srhmsubcALTV 47249 |
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