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Mirrors > Home > MPE Home > Th. List > rng1nfld | Structured version Visualization version GIF version |
Description: The zero ring is not a field. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
rng1nfld.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} |
Ref | Expression |
---|---|
rng1nfld | ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng1nfld.m | . . . . . 6 ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} | |
2 | 1 | rng1nnzr 19724 | . . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) |
3 | df-nel 3089 | . . . . 5 ⊢ (𝑀 ∉ NzRing ↔ ¬ 𝑀 ∈ NzRing) | |
4 | 2, 3 | sylib 219 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → ¬ 𝑀 ∈ NzRing) |
5 | drngnzr 19712 | . . . 4 ⊢ (𝑀 ∈ DivRing → 𝑀 ∈ NzRing) | |
6 | 4, 5 | nsyl 142 | . . 3 ⊢ (𝑍 ∈ 𝑉 → ¬ 𝑀 ∈ DivRing) |
7 | isfld 19189 | . . . 4 ⊢ (𝑀 ∈ Field ↔ (𝑀 ∈ DivRing ∧ 𝑀 ∈ CRing)) | |
8 | simpl 483 | . . . . 5 ⊢ ((𝑀 ∈ DivRing ∧ 𝑀 ∈ CRing) → 𝑀 ∈ DivRing) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → ((𝑀 ∈ DivRing ∧ 𝑀 ∈ CRing) → 𝑀 ∈ DivRing)) |
10 | 7, 9 | syl5bi 243 | . . 3 ⊢ (𝑍 ∈ 𝑉 → (𝑀 ∈ Field → 𝑀 ∈ DivRing)) |
11 | 6, 10 | mtod 199 | . 2 ⊢ (𝑍 ∈ 𝑉 → ¬ 𝑀 ∈ Field) |
12 | df-nel 3089 | . 2 ⊢ (𝑀 ∉ Field ↔ ¬ 𝑀 ∈ Field) | |
13 | 11, 12 | sylibr 235 | 1 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∉ wnel 3088 {csn 4466 {ctp 4470 〈cop 4472 ‘cfv 6217 ndxcnx 16297 Basecbs 16300 +gcplusg 16382 .rcmulr 16383 CRingccrg 18976 DivRingcdr 19180 Fieldcfield 19181 NzRingcnzr 19707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-dju 9165 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-xnn0 11805 df-z 11819 df-uz 12083 df-fz 12732 df-hash 13529 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-plusg 16395 df-mulr 16396 df-0g 16532 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-grp 17852 df-minusg 17853 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-drng 19182 df-field 19183 df-nzr 19708 |
This theorem is referenced by: (None) |
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