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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 2622 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2622 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 19259 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | isnzr2.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 4, 1 | ringidcl 18568 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
6 | 5 | adantr 481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ∈ 𝐵) |
7 | 4, 2 | ring0cl 18569 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ 𝐵) |
8 | 7 | adantr 481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (0g‘𝑅) ∈ 𝐵) |
9 | simpr 477 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ≠ (0g‘𝑅)) | |
10 | df-ne 2795 | . . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
11 | neeq1 2856 | . . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 ≠ 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) | |
12 | 10, 11 | syl5bbr 274 | . . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (¬ 𝑥 = 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) |
13 | neeq2 2857 | . . . . . . . . 9 ⊢ (𝑦 = (0g‘𝑅) → ((1r‘𝑅) ≠ 𝑦 ↔ (1r‘𝑅) ≠ (0g‘𝑅))) | |
14 | 12, 13 | rspc2ev 3324 | . . . . . . . 8 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵 ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
15 | 6, 8, 9, 14 | syl3anc 1326 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
16 | 15 | ex 450 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
17 | 4, 1, 2 | ring1eq0 18590 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
18 | 17 | 3expb 1266 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
19 | 18 | necon3bd 2808 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
20 | 19 | rexlimdvva 3038 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
21 | 16, 20 | impbid 202 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
22 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
23 | 4, 22 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
24 | 1sdom 8163 | . . . . . 6 ⊢ (𝐵 ∈ V → (1𝑜 ≺ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) | |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (1𝑜 ≺ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
26 | 21, 25 | syl6bbr 278 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 1𝑜 ≺ 𝐵)) |
27 | 1onn 7719 | . . . . . 6 ⊢ 1𝑜 ∈ ω | |
28 | sucdom 8157 | . . . . . 6 ⊢ (1𝑜 ∈ ω → (1𝑜 ≺ 𝐵 ↔ suc 1𝑜 ≼ 𝐵)) | |
29 | 27, 28 | ax-mp 5 | . . . . 5 ⊢ (1𝑜 ≺ 𝐵 ↔ suc 1𝑜 ≼ 𝐵) |
30 | df-2o 7561 | . . . . . 6 ⊢ 2𝑜 = suc 1𝑜 | |
31 | 30 | breq1i 4660 | . . . . 5 ⊢ (2𝑜 ≼ 𝐵 ↔ suc 1𝑜 ≼ 𝐵) |
32 | 29, 31 | bitr4i 267 | . . . 4 ⊢ (1𝑜 ≺ 𝐵 ↔ 2𝑜 ≼ 𝐵) |
33 | 26, 32 | syl6bb 276 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 2𝑜 ≼ 𝐵)) |
34 | 33 | pm5.32i 669 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ↔ (𝑅 ∈ Ring ∧ 2𝑜 ≼ 𝐵)) |
35 | 3, 34 | bitri 264 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2𝑜 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 class class class wbr 4653 suc csuc 5725 ‘cfv 5888 ωcom 7065 1𝑜c1o 7553 2𝑜c2o 7554 ≼ cdom 7953 ≺ csdm 7954 Basecbs 15857 0gc0g 16100 1rcur 18501 Ringcrg 18547 NzRingcnzr 19257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-nzr 19258 |
This theorem is referenced by: opprnzr 19265 znfld 19909 znidomb 19910 |
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