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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 20615 | . 2 ⊢ ℤring ∈ Ring | |
2 | eleq1 2899 | . . . 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 2820 | . . . . . . 7 ⊢ inf((𝑥 ∩ ℕ), ℝ, < ) = inf((𝑥 ∩ ℕ), ℝ, < ) | |
6 | 3, 4, 5 | zringlpirlem2 20627 | . . . . . 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 20628 | . . . . . . 7 ⊢ (((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) ∧ 𝑧 ∈ 𝑥) → inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧) |
11 | 10 | ralrimiva 3181 | . . . . . 6 ⊢ ((𝑥 ∈ (LIdeal‘ℤring) ∧ 𝑥 ≠ {0}) → ∀𝑧 ∈ 𝑥 inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧) |
12 | breq1 5062 | . . . . . . . 8 ⊢ (𝑦 = inf((𝑥 ∩ ℕ), ℝ, < ) → (𝑦 ∥ 𝑧 ↔ inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧)) | |
13 | 12 | ralbidv 3196 | . . . . . . 7 ⊢ (𝑦 = inf((𝑥 ∩ ℕ), ℝ, < ) → (∀𝑧 ∈ 𝑥 𝑦 ∥ 𝑧 ↔ ∀𝑧 ∈ 𝑥 inf((𝑥 ∩ ℕ), ℝ, < ) ∥ 𝑧)) |
14 | 13 | rspcev 3620 | . . . . . 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 2820 | . . . . . . . 8 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
17 | eqid 2820 | . . . . . . . 8 ⊢ (LPIdeal‘ℤring) = (LPIdeal‘ℤring) | |
18 | dvdsrzring 20625 | . . . . . . . 8 ⊢ ∥ = (∥r‘ℤring) | |
19 | 16, 17, 18 | lpigen 20024 | . . . . . . 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 20622 | . . . . . 6 ⊢ 0 = (0g‘ℤring) | |
24 | 17, 23 | lpi0 20015 | . . . . 5 ⊢ (ℤring ∈ Ring → {0} ∈ (LPIdeal‘ℤring)) |
25 | 1, 24 | mp1i 13 | . . . 4 ⊢ (𝑥 ∈ (LIdeal‘ℤring) → {0} ∈ (LPIdeal‘ℤring)) |
26 | 2, 22, 25 | pm2.61ne 3101 | . . 3 ⊢ (𝑥 ∈ (LIdeal‘ℤring) → 𝑥 ∈ (LPIdeal‘ℤring)) |
27 | 26 | ssriv 3964 | . 2 ⊢ (LIdeal‘ℤring) ⊆ (LPIdeal‘ℤring) |
28 | 17, 16 | islpir2 20019 | . 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 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 ∩ cin 3928 ⊆ wss 3929 {csn 4560 class class class wbr 5059 ‘cfv 6348 infcinf 8898 ℝcr 10529 0cc0 10530 < clt 10668 ℕcn 11631 ∥ cdvds 15602 Ringcrg 19292 LIdealclidl 19937 LPIdealclpidl 20009 LPIRclpir 20010 ℤringzring 20612 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
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 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-dvds 15603 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-cmn 18903 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-dvdsr 19386 df-subrg 19528 df-lmod 19631 df-lss 19699 df-lsp 19739 df-sra 19939 df-rgmod 19940 df-lidl 19941 df-rsp 19942 df-lpidl 20011 df-lpir 20012 df-cnfld 20541 df-zring 20613 |
This theorem is referenced by: (None) |
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