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Mirrors > Home > MPE Home > Th. List > ringlz | Structured version Visualization version GIF version |
Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
rngz.b | ⊢ 𝐵 = (Base‘𝑅) |
rngz.t | ⊢ · = (.r‘𝑅) |
rngz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ringlz | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 19295 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | rngz.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rngz.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 18123 | . . . . . 6 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2798 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 18125 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 1, 4, 6 | syl2anc2 588 | . . . . 5 ⊢ (𝑅 ∈ Ring → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 7 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | oveq1d 7150 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 0 (+g‘𝑅) 0 ) · 𝑋) = ( 0 · 𝑋)) |
10 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
12 | simpr 488 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | 11, 11, 12 | 3jca 1125 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
14 | rngz.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
15 | 2, 5, 14 | ringdir 19313 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ( 0 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (( 0 (+g‘𝑅) 0 ) · 𝑋) = (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋))) |
16 | 13, 15 | syldan 594 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 0 (+g‘𝑅) 0 ) · 𝑋) = (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋))) |
17 | 1 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
18 | simpl 486 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring) | |
19 | 2, 14 | ringcl 19307 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) ∈ 𝐵) |
20 | 18, 11, 12, 19 | syl3anc 1368 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) ∈ 𝐵) |
21 | 2, 5, 3 | grprid 18126 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ( 0 · 𝑋) ∈ 𝐵) → (( 0 · 𝑋)(+g‘𝑅) 0 ) = ( 0 · 𝑋)) |
22 | 21 | eqcomd 2804 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ ( 0 · 𝑋) ∈ 𝐵) → ( 0 · 𝑋) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
23 | 17, 20, 22 | syl2anc 587 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
24 | 9, 16, 23 | 3eqtr3d 2841 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋)) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
25 | 2, 5 | grplcan 18153 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (( 0 · 𝑋) ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ ( 0 · 𝑋) ∈ 𝐵)) → ((( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋)) = (( 0 · 𝑋)(+g‘𝑅) 0 ) ↔ ( 0 · 𝑋) = 0 )) |
26 | 17, 20, 11, 20, 25 | syl13anc 1369 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋)) = (( 0 · 𝑋)(+g‘𝑅) 0 ) ↔ ( 0 · 𝑋) = 0 )) |
27 | 24, 26 | mpbid 235 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Grpcgrp 18095 Ringcrg 19290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19233 df-ring 19292 |
This theorem is referenced by: ringsrg 19335 ring1eq0 19336 ringnegl 19340 mulgass2 19347 gsumdixp 19355 dvdsr01 19401 0unit 19426 irredn0 19449 drngmul0or 19516 cntzsubr 19561 cntzsdrg 19574 isabvd 19584 domneq0 20063 frlmphllem 20469 psrlidm 20641 mplsubrglem 20677 mplmonmul 20704 evlslem4 20747 evlslem3 20752 evlslem6 20753 coe1tmmul 20906 cply1mul 20923 mamulid 21046 dmatmul 21102 scmatscm 21118 1mavmul 21153 mdetdiaglem 21203 mdetr0 21210 mdegmullem 24679 coe1mul3 24700 fta1glem1 24766 dvdschrmulg 30908 rmfsupp2 30917 elrspunidl 31014 fedgmullem1 31113 lflsc0N 36379 hdmapinvlem3 39216 hdmapinvlem4 39217 mnringmulrcld 40936 zrrnghm 44541 zlidlring 44552 rmsupp0 44770 ply1mulgsumlem2 44795 |
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