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Mirrors > Home > MPE Home > Th. List > zringlpir | Structured version Visualization version GIF version |
Description: The integers are a principal ideal ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.) |
Ref | Expression |
---|---|
zringlpir | ⊢ ℤring ∈ LPIR |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringring 20620 | . 2 ⊢ ℤring ∈ Ring | |
2 | eleq1 2900 | . . . 4 ⊢ (𝑥 = {0} → (𝑥 ∈ (LPIdeal‘ℤring) ↔ {0} ∈ (LPIdeal‘ℤring))) | |
3 | simpl 485 | . . . . . . 7 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → 𝑥 ∈ (LIdeal‘ℤring)) | |
4 | simpr 487 | . . . . . . 7 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → 𝑥 ≠ {0}) | |
5 | eqid 2821 | . . . . . . 7 ⊢ inf((𝑥 ∩ ℕ), ℝ, < ) = inf((𝑥 ∩ ℕ), ℝ, < ) | |
6 | 3, 4, 5 | zringlpirlem2 20632 | . . . . . 6 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → inf((𝑥 ∩ ℕ), ℝ, < ) ∈ 𝑥) |
7 | simpll 765 | . . . . . . . 8 ⊢ (((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (LIdeal‘ℤring)) | |
8 | simplr 767 | . . . . . . . 8 ⊢ (((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) ∧ 𝑧 ∈ 𝑥) → 𝑥 ≠ {0}) | |
9 | simpr 487 | . . . . . . . 8 ⊢ (((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥) | |
10 | 7, 8, 5, 9 | zringlpirlem3 20633 | . . . . . . 7 ⊢ (((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) ∧ 𝑧 ∈ 𝑥) → inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧) |
11 | 10 | ralrimiva 3182 | . . . . . 6 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → ∀𝑧 ∈ 𝑥 inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧) |
12 | breq1 5069 | . . . . . . . 8 ⊢ (𝑦 = inf((𝑥 ∩ ℕ), ℝ, < ) → (𝑦 ∥ 𝑧 ↔ inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧)) | |
13 | 12 | ralbidv 3197 | . . . . . . 7 ⊢ (𝑦 = inf((𝑥 ∩ ℕ), ℝ, < ) → (∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧 ↔ ∀𝑧 ∈ 𝑥 inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧)) |
14 | 13 | rspcev 3623 | . . . . . 6 ⊢ ((inf((𝑥 ∩ ℕ), ℝ, < ) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧) |
15 | 6, 11, 14 | syl2anc 586 | . . . . 5 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧) |
16 | eqid 2821 | . . . . . . . 8 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
17 | eqid 2821 | . . . . . . . 8 ⊢ (LPIdeal‘ℤring) = (LPIdeal‘ℤring) | |
18 | dvdsrzring 20630 | . . . . . . . 8 ⊢ ∥ = (∥r‘ℤring) | |
19 | 16, 17, 18 | lpigen 20029 | . . . . . . 7 ⊢ ((ℤring ∈ Ring ∧ 𝑥 ∈ (LIdeal‘ℤring)) → (𝑥 ∈ (LPIdeal‘ℤring) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧)) |
20 | 1, 19 | mpan 688 | . . . . . 6 ⊢ (𝑥 ∈ (LIdeal‘ℤring) → (𝑥 ∈ (LPIdeal‘ℤring) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧)) |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → (𝑥 ∈ (LPIdeal‘ℤring) ↔ ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧)) |
22 | 15, 21 | mpbird 259 | . . . 4 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → 𝑥 ∈ (LPIdeal‘ℤring)) |
23 | zring0 20627 | . . . . . 6 ⊢ 0 = (0g‘ℤring) | |
24 | 17, 23 | lpi0 20020 | . . . . 5 ⊢ (ℤring ∈ Ring → {0} ∈ (LPIdeal‘ℤring)) |
25 | 1, 24 | mp1i 13 | . . . 4 ⊢ (𝑥 ∈ (LIdeal‘ℤring) → {0} ∈ (LPIdeal‘ℤring)) |
26 | 2, 22, 25 | pm2.61ne 3102 | . . 3 ⊢ (𝑥 ∈ (LIdeal‘ℤring) → 𝑥 ∈ (LPIdeal‘ℤring)) |
27 | 26 | ssriv 3971 | . 2 ⊢ (LIdeal‘ℤring) ⊆ (LPIdeal‘ℤring) |
28 | 17, 16 | islpir2 20024 | . 2 ⊢ (ℤring ∈ LPIR ↔ (ℤring ∈ Ring ∧ (LIdeal‘ℤring) ⊆ (LPIdeal‘ℤring))) |
29 | 1, 27, 28 | mpbir2an 709 | 1 ⊢ ℤring ∈ LPIR |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ‘cfv 6355 infcinf 8905 ℝcr 10536 0cc0 10537 < clt 10675 ℕcn 11638 ∥ cdvds 15607 Ringcrg 19297 LIdealclidl 19942 LPIdealclpidl 20014 LPIRclpir 20015 ℤringzring 20617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-dvdsr 19391 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-lpidl 20016 df-lpir 20017 df-cnfld 20546 df-zring 20618 |
This theorem is referenced by: (None) |
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