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Mirrors > Home > MPE Home > Th. List > srng0 | Structured version Visualization version GIF version |
Description: The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
srng0.i | ⊢ ∗ = (*𝑟‘𝑅) |
srng0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
srng0 | ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngring 20821 | . . 3 ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | |
2 | ringgrp 20217 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2726 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | srng0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | grpidcl 18955 | . . 3 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) |
6 | srng0.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
7 | eqid 2726 | . . . 4 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
8 | 3, 6, 7 | stafval 20817 | . . 3 ⊢ ( 0 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘ 0 ) = ( ∗ ‘ 0 )) |
9 | 1, 2, 5, 8 | 4syl 19 | . 2 ⊢ (𝑅 ∈ *-Ring → ((*rf‘𝑅)‘ 0 ) = ( ∗ ‘ 0 )) |
10 | eqid 2726 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
11 | 10, 7 | srngrhm 20820 | . . 3 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅))) |
12 | rhmghm 20462 | . . 3 ⊢ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) → (*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅))) | |
13 | 10, 4 | oppr0 20327 | . . . 4 ⊢ 0 = (0g‘(oppr‘𝑅)) |
14 | 4, 13 | ghmid 19212 | . . 3 ⊢ ((*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅)) → ((*rf‘𝑅)‘ 0 ) = 0 ) |
15 | 11, 12, 14 | 3syl 18 | . 2 ⊢ (𝑅 ∈ *-Ring → ((*rf‘𝑅)‘ 0 ) = 0 ) |
16 | 9, 15 | eqtr3d 2768 | 1 ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 *𝑟cstv 17263 0gc0g 17449 Grpcgrp 18923 GrpHom cghm 19202 Ringcrg 20212 opprcoppr 20311 RingHom crh 20447 *rfcstf 20812 *-Ringcsr 20813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mulr 17275 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-ghm 19203 df-mgp 20114 df-ur 20161 df-ring 20214 df-oppr 20312 df-rhm 20450 df-staf 20814 df-srng 20815 |
This theorem is referenced by: iporthcom 21627 ip0r 21629 |
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