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Mirrors > Home > MPE Home > Th. List > rngrz | Structured version Visualization version GIF version |
Description: The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 20319. (Revised by AV, 16-Feb-2025.) |
Ref | Expression |
---|---|
rngcl.b | ⊢ 𝐵 = (Base‘𝑅) |
rngcl.t | ⊢ · = (.r‘𝑅) |
rnglz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rngrz | ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnggrp 20187 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
2 | rngcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rnglz.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 19007 | . . . . . 6 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2740 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 19009 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 1, 4, 6 | syl2anc2 584 | . . . . 5 ⊢ (𝑅 ∈ Rng → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 7 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | oveq2d 7466 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = (𝑋 · 0 )) |
10 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
11 | 2, 3 | rng0cl 20192 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
13 | 10, 12, 12 | 3jca 1128 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) |
14 | rngcl.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
15 | 2, 5, 14 | rngdi 20189 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
16 | 13, 15 | syldan 590 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
17 | 1 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
18 | 2, 14 | rngcl 20193 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
19 | 12, 18 | mpd3an3 1462 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
20 | 2, 5, 3 | grplid 19009 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 0 )) = (𝑋 · 0 )) |
21 | 20 | eqcomd 2746 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
22 | 17, 19, 21 | syl2anc 583 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
23 | 9, 16, 22 | 3eqtr3d 2788 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
24 | 2, 5 | grprcan 19015 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ ((𝑋 · 0 ) ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ (𝑋 · 0 ) ∈ 𝐵)) → (((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 )) ↔ (𝑋 · 0 ) = 0 )) |
25 | 17, 19, 12, 19, 24 | syl13anc 1372 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 )) ↔ (𝑋 · 0 ) = 0 )) |
26 | 23, 25 | mpbid 232 | 1 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 +gcplusg 17313 .rcmulr 17314 0gc0g 17501 Grpcgrp 18975 Rngcrng 20181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-plusg 17326 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-grp 18978 df-abl 19827 df-mgp 20164 df-rng 20182 |
This theorem is referenced by: rngmneg2 20197 ringrz 20319 cntzsubrng 20595 rnglidl0 21264 |
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