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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 2820 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2820 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 20028 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | isnzr2.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 4, 1 | ringidcl 19314 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
6 | 5 | adantr 483 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ∈ 𝐵) |
7 | 4, 2 | ring0cl 19315 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ 𝐵) |
8 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (0g‘𝑅) ∈ 𝐵) |
9 | simpr 487 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (1r‘𝑅) ≠ (0g‘𝑅)) | |
10 | df-ne 3016 | . . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
11 | neeq1 3077 | . . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 ≠ 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) | |
12 | 10, 11 | syl5bbr 287 | . . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (¬ 𝑥 = 𝑦 ↔ (1r‘𝑅) ≠ 𝑦)) |
13 | neeq2 3078 | . . . . . . . . 9 ⊢ (𝑦 = (0g‘𝑅) → ((1r‘𝑅) ≠ 𝑦 ↔ (1r‘𝑅) ≠ (0g‘𝑅))) | |
14 | 12, 13 | rspc2ev 3634 | . . . . . . . 8 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵 ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
15 | 6, 8, 9, 14 | syl3anc 1366 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦) |
16 | 15 | ex 415 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
17 | 4, 1, 2 | ring1eq0 19336 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
18 | 17 | 3expb 1115 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((1r‘𝑅) = (0g‘𝑅) → 𝑥 = 𝑦)) |
19 | 18 | necon3bd 3029 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
20 | 19 | rexlimdvva 3293 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦 → (1r‘𝑅) ≠ (0g‘𝑅))) |
21 | 16, 20 | impbid 214 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ 𝑥 = 𝑦)) |
22 | 4 | fvexi 6681 | . . . . . 6 ⊢ 𝐵 ∈ V |
23 | 1sdom 8718 | . . . . . 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 | syl6bbr 291 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 1o ≺ 𝐵)) |
26 | 1onn 8262 | . . . . . 6 ⊢ 1o ∈ ω | |
27 | sucdom 8712 | . . . . . 6 ⊢ (1o ∈ ω → (1o ≺ 𝐵 ↔ suc 1o ≼ 𝐵)) | |
28 | 26, 27 | ax-mp 5 | . . . . 5 ⊢ (1o ≺ 𝐵 ↔ suc 1o ≼ 𝐵) |
29 | df-2o 8100 | . . . . . 6 ⊢ 2o = suc 1o | |
30 | 29 | breq1i 5070 | . . . . 5 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵) |
31 | 28, 30 | bitr4i 280 | . . . 4 ⊢ (1o ≺ 𝐵 ↔ 2o ≼ 𝐵) |
32 | 25, 31 | syl6bb 289 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ 2o ≼ 𝐵)) |
33 | 32 | pm5.32i 577 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵)) |
34 | 3, 33 | bitri 277 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 Vcvv 3493 class class class wbr 5063 suc csuc 6190 ‘cfv 6352 ωcom 7577 1oc1o 8092 2oc2o 8093 ≼ cdom 8504 ≺ csdm 8505 Basecbs 16479 0gc0g 16709 1rcur 19247 Ringcrg 19293 NzRingcnzr 20026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-plusg 16574 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-grp 18102 df-minusg 18103 df-mgp 19236 df-ur 19248 df-ring 19295 df-nzr 20027 |
This theorem is referenced by: opprnzr 20034 znfld 20703 znidomb 20704 |
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