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Mirrors > Home > MPE Home > Th. List > isnzr2hash | Structured version Visualization version GIF version |
Description: Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 19311. (Contributed by AV, 14-Apr-2019.) |
Ref | Expression |
---|---|
isnzr2hash.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
isnzr2hash | ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2651 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 19307 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | isnzr2hash.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 4, 1 | ringidcl 18614 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
6 | 4, 2 | ring0cl 18615 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ 𝐵) |
7 | 1re 10077 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | 7 | rexri 10135 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 ∈ ℝ*) |
10 | prex 4939 | . . . . . . . 8 ⊢ {(1r‘𝑅), (0g‘𝑅)} ∈ V | |
11 | hashxrcl 13186 | . . . . . . . 8 ⊢ ({(1r‘𝑅), (0g‘𝑅)} ∈ V → (#‘{(1r‘𝑅), (0g‘𝑅)}) ∈ ℝ*) | |
12 | 10, 11 | mp1i 13 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (#‘{(1r‘𝑅), (0g‘𝑅)}) ∈ ℝ*) |
13 | fvex 6239 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
14 | 4, 13 | eqeltri 2726 | . . . . . . . 8 ⊢ 𝐵 ∈ V |
15 | hashxrcl 13186 | . . . . . . . 8 ⊢ (𝐵 ∈ V → (#‘𝐵) ∈ ℝ*) | |
16 | 14, 15 | mp1i 13 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (#‘𝐵) ∈ ℝ*) |
17 | 1lt2 11232 | . . . . . . . 8 ⊢ 1 < 2 | |
18 | hashprg 13220 | . . . . . . . . 9 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ (#‘{(1r‘𝑅), (0g‘𝑅)}) = 2)) | |
19 | 18 | biimpa 500 | . . . . . . . 8 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (#‘{(1r‘𝑅), (0g‘𝑅)}) = 2) |
20 | 17, 19 | syl5breqr 4723 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 < (#‘{(1r‘𝑅), (0g‘𝑅)})) |
21 | simpl 472 | . . . . . . . . 9 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵)) | |
22 | fvex 6239 | . . . . . . . . . 10 ⊢ (1r‘𝑅) ∈ V | |
23 | fvex 6239 | . . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V | |
24 | 22, 23 | prss 4383 | . . . . . . . . 9 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ↔ {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) |
25 | 21, 24 | sylib 208 | . . . . . . . 8 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) |
26 | hashss 13235 | . . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) → (#‘{(1r‘𝑅), (0g‘𝑅)}) ≤ (#‘𝐵)) | |
27 | 14, 25, 26 | sylancr 696 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (#‘{(1r‘𝑅), (0g‘𝑅)}) ≤ (#‘𝐵)) |
28 | 9, 12, 16, 20, 27 | xrltletrd 12030 | . . . . . 6 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 < (#‘𝐵)) |
29 | 28 | ex 449 | . . . . 5 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) → ((1r‘𝑅) ≠ (0g‘𝑅) → 1 < (#‘𝐵))) |
30 | 5, 6, 29 | syl2anc 694 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) → 1 < (#‘𝐵))) |
31 | 30 | imdistani 726 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝑅 ∈ Ring ∧ 1 < (#‘𝐵))) |
32 | simpl 472 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 𝑅 ∈ Ring) | |
33 | 4, 1, 2 | ring1ne0 18637 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → (1r‘𝑅) ≠ (0g‘𝑅)) |
34 | 32, 33 | jca 553 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
35 | 31, 34 | impbii 199 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵))) |
36 | 3, 35 | bitri 264 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 ⊆ wss 3607 {cpr 4212 class class class wbr 4685 ‘cfv 5926 1c1 9975 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 2c2 11108 #chash 13157 Basecbs 15904 0gc0g 16147 1rcur 18547 Ringcrg 18593 NzRingcnzr 19305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-fz 12365 df-hash 13158 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-mgp 18536 df-ur 18548 df-ring 18595 df-nzr 19306 |
This theorem is referenced by: 0ringnnzr 19317 el0ldepsnzr 42581 |
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