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Mirrors > Home > MPE Home > Th. List > isnzr2 | Structured version Visualization version GIF version |
Description: Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
isnzr2.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
isnzr2 | ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2737 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 20652 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | isnzr2.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 4, 1 | ringidcl 19913 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
6 | 5 | adantr 481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ∈ 𝐵) |
7 | 4, 2 | ring0cl 19914 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ 𝐵) |
8 | 7 | adantr 481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (0g‘𝑅) ∈ 𝐵) |
9 | simpr 485 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ≠ (0g‘𝑅)) | |
10 | df-ne 2942 | . . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
11 | neeq1 3004 | . . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 ≠ 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) | |
12 | 10, 11 | bitr3id 284 | . . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (¬ 𝑥 = 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) |
13 | neeq2 3005 | . . . . . . . . 9 ⊢ (𝑦 = (0g‘𝑅) → ((1r‘𝑅) ≠ 𝑦 ↔ (1r‘𝑅) ≠ (0g‘𝑅))) | |
14 | 12, 13 | rspc2ev 3590 | . . . . . . . 8 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵 ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
15 | 6, 8, 9, 14 | syl3anc 1371 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
16 | 15 | ex 413 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
17 | 4, 1, 2 | ring1eq0 19937 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
18 | 17 | 3expb 1120 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
19 | 18 | necon3bd 2955 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
20 | 19 | rexlimdvva 3203 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
21 | 16, 20 | impbid 211 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
22 | 4 | fvexi 6851 | . . . . . 6 ⊢ 𝐵 ∈ V |
23 | 1sdom 9125 | . . . . . 6 ⊢ (𝐵 ∈ V → (1o ≺ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) | |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (1o ≺ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
25 | 21, 24 | bitr4di 288 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 1o ≺ 𝐵)) |
26 | 1onn 8553 | . . . . . 6 ⊢ 1o ∈ ω | |
27 | sucdom 9112 | . . . . . 6 ⊢ (1o ∈ ω → (1o ≺ 𝐵 ↔ suc 1o ≼ 𝐵)) | |
28 | 26, 27 | ax-mp 5 | . . . . 5 ⊢ (1o ≺ 𝐵 ↔ suc 1o ≼ 𝐵) |
29 | df-2o 8380 | . . . . . 6 ⊢ 2o = suc 1o | |
30 | 29 | breq1i 5110 | . . . . 5 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵) |
31 | 28, 30 | bitr4i 277 | . . . 4 ⊢ (1o ≺ 𝐵 ↔ 2o ≼ 𝐵) |
32 | 25, 31 | bitrdi 286 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 2o ≼ 𝐵)) |
33 | 32 | pm5.32i 575 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵)) |
34 | 3, 33 | bitri 274 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 Vcvv 3443 class class class wbr 5103 suc csuc 6315 ‘cfv 6491 ωcom 7792 1oc1o 8372 2oc2o 8373 ≼ cdom 8814 ≺ csdm 8815 Basecbs 17017 0gc0g 17255 1rcur 19842 Ringcrg 19888 NzRingcnzr 20650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-plusg 17080 df-0g 17257 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-grp 18685 df-minusg 18686 df-mgp 19826 df-ur 19843 df-ring 19890 df-nzr 20651 |
This theorem is referenced by: opprnzr 20658 znfld 20890 znidomb 20891 |
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