Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19304 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | rngz.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rngz.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 18133 | . . . . . 6 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2823 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 18135 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 1, 4, 6 | syl2anc2 587 | . . . . 5 ⊢ (𝑅 ∈ Ring → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | oveq1d 7173 | . . 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 483 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
12 | simpr 487 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | 11, 11, 12 | 3jca 1124 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
14 | rngz.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
15 | 2, 5, 14 | ringdir 19319 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ( 0 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (( 0 (+g‘𝑅) 0 ) · 𝑋) = (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋))) |
16 | 13, 15 | syldan 593 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 0 (+g‘𝑅) 0 ) · 𝑋) = (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋))) |
17 | 1 | adantr 483 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
18 | simpl 485 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring) | |
19 | 2, 14 | ringcl 19313 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) ∈ 𝐵) |
20 | 18, 11, 12, 19 | syl3anc 1367 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) ∈ 𝐵) |
21 | 2, 5, 3 | grprid 18136 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ( 0 · 𝑋) ∈ 𝐵) → (( 0 · 𝑋)(+g‘𝑅) 0 ) = ( 0 · 𝑋)) |
22 | 21 | eqcomd 2829 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ ( 0 · 𝑋) ∈ 𝐵) → ( 0 · 𝑋) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
23 | 17, 20, 22 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
24 | 9, 16, 23 | 3eqtr3d 2866 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋)) = (( 0 · 𝑋)(+g‘𝑅) 0 )) |
25 | 2, 5 | grplcan 18163 | . . 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 1368 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((( 0 · 𝑋)(+g‘𝑅)( 0 · 𝑋)) = (( 0 · 𝑋)(+g‘𝑅) 0 ) ↔ ( 0 · 𝑋) = 0 )) |
27 | 24, 26 | mpbid 234 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 0gc0g 16715 Grpcgrp 18105 Ringcrg 19299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ring 19301 |
This theorem is referenced by: ringsrg 19341 ring1eq0 19342 ringnegl 19346 mulgass2 19353 gsumdixp 19361 dvdsr01 19407 0unit 19432 irredn0 19455 drngmul0or 19525 cntzsubr 19570 cntzsdrg 19583 isabvd 19593 domneq0 20072 psrlidm 20185 mplsubrglem 20221 mplmonmul 20247 evlslem4 20290 evlslem3 20295 evlslem6 20296 coe1tmmul 20447 cply1mul 20464 frlmphllem 20926 mamulid 21052 dmatmul 21108 scmatscm 21124 1mavmul 21159 mdetdiaglem 21209 mdetr0 21216 mdegmullem 24674 coe1mul3 24695 fta1glem1 24761 dvdschrmulg 30860 rmfsupp2 30868 fedgmullem1 31027 lflsc0N 36221 hdmapinvlem3 39058 hdmapinvlem4 39059 zrrnghm 44195 zlidlring 44206 rmsupp0 44423 ply1mulgsumlem2 44448 |
Copyright terms: Public domain | W3C validator |