Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19622 | . . 3 ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | |
2 | ringgrp 19301 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | srng0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | grpidcl 18130 | . . 3 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) |
6 | srng0.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
7 | eqid 2821 | . . . 4 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
8 | 3, 6, 7 | stafval 19618 | . . 3 ⊢ ( 0 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘ 0 ) = ( ∗ ‘ 0 )) |
9 | 1, 2, 5, 8 | 4syl 19 | . 2 ⊢ (𝑅 ∈ *-Ring → ((*rf‘𝑅)‘ 0 ) = ( ∗ ‘ 0 )) |
10 | eqid 2821 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
11 | 10, 7 | srngrhm 19621 | . . 3 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅))) |
12 | rhmghm 19476 | . . 3 ⊢ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) → (*rf‘𝑅) ∈ (𝑅 GrpHom (oppr‘𝑅))) | |
13 | 10, 4 | oppr0 19382 | . . . 4 ⊢ 0 = (0g‘(oppr‘𝑅)) |
14 | 4, 13 | ghmid 18363 | . . 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 2858 | 1 ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 *𝑟cstv 16566 0gc0g 16712 Grpcgrp 18102 GrpHom cghm 18354 Ringcrg 19296 opprcoppr 19371 RingHom crh 19463 *rfcstf 19613 *-Ringcsr 19614 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-mulr 16578 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-grp 18105 df-ghm 18355 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-rnghom 19466 df-staf 19615 df-srng 19616 |
This theorem is referenced by: iporthcom 20778 ip0r 20780 |
Copyright terms: Public domain | W3C validator |