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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 19970 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | idrhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 3 | idghm 19024 | . . . 4 ⊢ (𝑅 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅)) |
6 | eqid 2737 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6 | ringmgp 19971 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
8 | 6, 3 | mgpbas 19903 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
9 | 8 | idmhm 18612 | . . . 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 513 | . 2 ⊢ (𝑅 ∈ Ring → (( I ↾ 𝐵) ∈ (𝑅 GrpHom 𝑅) ∧ ( I ↾ 𝐵) ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑅)))) |
12 | 6, 6 | isrhm 20153 | . 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 1345 | 1 ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 I cid 5531 ↾ cres 5636 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 Mndcmnd 18557 MndHom cmhm 18600 Grpcgrp 18749 GrpHom cghm 19006 mulGrpcmgp 19897 Ringcrg 19965 RingHom crh 20144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-ghm 19007 df-mgp 19898 df-ur 19915 df-ring 19967 df-rnghom 20147 |
This theorem is referenced by: rhmsubcsetclem1 46326 rhmsubcrngclem1 46332 funcringcsetcALTV2lem7 46347 ringccatidALTV 46357 funcringcsetclem7ALTV 46370 srhmsubc 46381 rhmsubclem3 46393 srhmsubcALTV 46399 rhmsubcALTVlem3 46411 |
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