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Mirrors > Home > ILE Home > Th. List > rrgnz | GIF version |
Description: In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.) |
Ref | Expression |
---|---|
rrgnz.t | ⊢ 𝐸 = (RLReg‘𝑅) |
rrgnz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rrgnz | ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2193 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | rrgnz.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 13678 | . . 3 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
4 | 3 | neneqd 2385 | . 2 ⊢ (𝑅 ∈ NzRing → ¬ (1r‘𝑅) = 0 ) |
5 | nzrring 13679 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
6 | 5 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 0 ∈ 𝐸) → 𝑅 ∈ Ring) |
7 | simpr 110 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 0 ∈ 𝐸) → 0 ∈ 𝐸) | |
8 | eqid 2193 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 1 | ringidcl 13516 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 6, 9 | syl 14 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 0 ∈ 𝐸) → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | eqid 2193 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 8, 11, 2, 6, 10 | ringlzd 13541 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 0 ∈ 𝐸) → ( 0 (.r‘𝑅)(1r‘𝑅)) = 0 ) |
13 | rrgnz.t | . . . . 5 ⊢ 𝐸 = (RLReg‘𝑅) | |
14 | 13, 8, 11, 2 | rrgeq0 13761 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝐸 ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (( 0 (.r‘𝑅)(1r‘𝑅)) = 0 ↔ (1r‘𝑅) = 0 )) |
15 | 14 | biimpa 296 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 0 ∈ 𝐸 ∧ (1r‘𝑅) ∈ (Base‘𝑅)) ∧ ( 0 (.r‘𝑅)(1r‘𝑅)) = 0 ) → (1r‘𝑅) = 0 ) |
16 | 6, 7, 10, 12, 15 | syl31anc 1252 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 0 ∈ 𝐸) → (1r‘𝑅) = 0 ) |
17 | 4, 16 | mtand 666 | 1 ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ‘cfv 5254 (class class class)co 5918 Basecbs 12618 .rcmulr 12696 0gc0g 12867 1rcur 13455 Ringcrg 13492 NzRingcnzr 13675 RLRegcrlreg 13751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-pre-ltirr 7984 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-ltxr 8059 df-inn 8983 df-2 9041 df-3 9042 df-ndx 12621 df-slot 12622 df-base 12624 df-sets 12625 df-plusg 12708 df-mulr 12709 df-0g 12869 df-mgm 12939 df-sgrp 12985 df-mnd 12998 df-grp 13075 df-minusg 13076 df-mgp 13417 df-ur 13456 df-ring 13494 df-nzr 13676 df-rlreg 13754 |
This theorem is referenced by: (None) |
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