![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20410. (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 2724 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2724 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 20406 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | isnzr2hash.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 4, 1 | ringidcl 20155 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
6 | 4, 2 | ring0cl 20156 | . . . . 5 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ 𝐵) |
7 | 1xr 11270 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 ∈ ℝ*) |
9 | prex 5422 | . . . . . . . 8 ⊢ {(1r‘𝑅), (0g‘𝑅)} ∈ V | |
10 | hashxrcl 14314 | . . . . . . . 8 ⊢ ({(1r‘𝑅), (0g‘𝑅)} ∈ V → (♯‘{(1r‘𝑅), (0g‘𝑅)}) ∈ ℝ*) | |
11 | 9, 10 | mp1i 13 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (♯‘{(1r‘𝑅), (0g‘𝑅)}) ∈ ℝ*) |
12 | 4 | fvexi 6895 | . . . . . . . 8 ⊢ 𝐵 ∈ V |
13 | hashxrcl 14314 | . . . . . . . 8 ⊢ (𝐵 ∈ V → (♯‘𝐵) ∈ ℝ*) | |
14 | 12, 13 | mp1i 13 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (♯‘𝐵) ∈ ℝ*) |
15 | 1lt2 12380 | . . . . . . . 8 ⊢ 1 < 2 | |
16 | hashprg 14352 | . . . . . . . . 9 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) → ((1r‘𝑅) ≠ (0g‘𝑅) ↔ (♯‘{(1r‘𝑅), (0g‘𝑅)}) = 2)) | |
17 | 16 | biimpa 476 | . . . . . . . 8 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (♯‘{(1r‘𝑅), (0g‘𝑅)}) = 2) |
18 | 15, 17 | breqtrrid 5176 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 < (♯‘{(1r‘𝑅), (0g‘𝑅)})) |
19 | simpl 482 | . . . . . . . . 9 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵)) | |
20 | fvex 6894 | . . . . . . . . . 10 ⊢ (1r‘𝑅) ∈ V | |
21 | fvex 6894 | . . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V | |
22 | 20, 21 | prss 4815 | . . . . . . . . 9 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ↔ {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) |
23 | 19, 22 | sylib 217 | . . . . . . . 8 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) |
24 | hashss 14366 | . . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ {(1r‘𝑅), (0g‘𝑅)} ⊆ 𝐵) → (♯‘{(1r‘𝑅), (0g‘𝑅)}) ≤ (♯‘𝐵)) | |
25 | 12, 23, 24 | sylancr 586 | . . . . . . 7 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (♯‘{(1r‘𝑅), (0g‘𝑅)}) ≤ (♯‘𝐵)) |
26 | 8, 11, 14, 18, 25 | xrltletrd 13137 | . . . . . 6 ⊢ ((((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → 1 < (♯‘𝐵)) |
27 | 26 | ex 412 | . . . . 5 ⊢ (((1r‘𝑅) ∈ 𝐵 ∧ (0g‘𝑅) ∈ 𝐵) → ((1r‘𝑅) ≠ (0g‘𝑅) → 1 < (♯‘𝐵))) |
28 | 5, 6, 27 | syl2anc 583 | . . . 4 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅) ≠ (0g‘𝑅) → 1 < (♯‘𝐵))) |
29 | 28 | imdistani 568 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵))) |
30 | simpl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 𝑅 ∈ Ring) | |
31 | 4, 1, 2 | ring1ne0 20188 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → (1r‘𝑅) ≠ (0g‘𝑅)) |
32 | 30, 31 | jca 511 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
33 | 29, 32 | impbii 208 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵))) |
34 | 3, 33 | bitri 275 | 1 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ⊆ wss 3940 {cpr 4622 class class class wbr 5138 ‘cfv 6533 1c1 11107 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 2c2 12264 ♯chash 14287 Basecbs 17143 0gc0g 17384 1rcur 20076 Ringcrg 20128 NzRingcnzr 20404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-xnn0 12542 df-z 12556 df-uz 12820 df-fz 13482 df-hash 14288 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-nzr 20405 |
This theorem is referenced by: 0ringnnzr 20415 prmidl0 33038 qsidomlem1 33040 krull 33063 el0ldepsnzr 47336 |
Copyright terms: Public domain | W3C validator |