![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20312. (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 20180 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
2 | rngcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rnglz.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 19000 | . . . . . 6 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2734 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 19002 | . . . . . 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 7461 | . . 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 20185 | . . . . . 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 20182 | . . . 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 20186 | . . . . 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 19002 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 0 )) = (𝑋 · 0 )) |
21 | 20 | eqcomd 2740 | . . . 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 2782 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
24 | 2, 5 | grprcan 19008 | . . 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 2103 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 +gcplusg 17306 .rcmulr 17307 0gc0g 17494 Grpcgrp 18968 Rngcrng 20174 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-grp 18971 df-abl 19820 df-mgp 20157 df-rng 20175 |
This theorem is referenced by: rngmneg2 20190 ringrz 20312 cntzsubrng 20588 rnglidl0 21257 |
Copyright terms: Public domain | W3C validator |