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Mirrors > Home > MPE Home > Th. List > zringlpirlem1 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 20182. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
zringlpirlem.i | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) |
zringlpirlem.n0 | ⊢ (𝜑 → 𝐼 ≠ {0}) |
Ref | Expression |
---|---|
zringlpirlem1 | ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 768 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → 𝑎 ∈ 𝐼) | |
2 | eleq1 2877 | . . . . . 6 ⊢ ((abs‘𝑎) = 𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ 𝑎 ∈ 𝐼)) | |
3 | 1, 2 | syl5ibrcom 250 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 → (abs‘𝑎) ∈ 𝐼)) |
4 | zsubrg 20144 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
5 | subrgsubg 19534 | . . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
7 | zringlpirlem.i | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) | |
8 | zringbas 20169 | . . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤring) | |
9 | eqid 2798 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
10 | 8, 9 | lidlss 19976 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ) |
12 | 11 | sselda 3915 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ) |
13 | df-zring 20164 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
14 | eqid 2798 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
15 | eqid 2798 | . . . . . . . . . . 11 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
16 | 13, 14, 15 | subginv 18278 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑎 ∈ ℤ) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
17 | 6, 12, 16 | sylancr 590 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
18 | 12 | zcnd 12076 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ) |
19 | cnfldneg 20117 | . . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((invg‘ℂfld)‘𝑎) = -𝑎) | |
20 | 18, 19 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = -𝑎) |
21 | 17, 20 | eqtr3d 2835 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) = -𝑎) |
22 | zringring 20166 | . . . . . . . . 9 ⊢ ℤring ∈ Ring | |
23 | 7 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘ℤring)) |
24 | simpr 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼) | |
25 | 9, 15 | lidlnegcl 19980 | . . . . . . . . 9 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
26 | 22, 23, 24, 25 | mp3an2i 1463 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
27 | 21, 26 | eqeltrrd 2891 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → -𝑎 ∈ 𝐼) |
28 | 27 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → -𝑎 ∈ 𝐼) |
29 | eleq1 2877 | . . . . . 6 ⊢ ((abs‘𝑎) = -𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ -𝑎 ∈ 𝐼)) | |
30 | 28, 29 | syl5ibrcom 250 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = -𝑎 → (abs‘𝑎) ∈ 𝐼)) |
31 | 12 | zred 12075 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ) |
32 | 31 | absord 14767 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
33 | 32 | adantr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
34 | 3, 30, 33 | mpjaod 857 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ 𝐼) |
35 | nnabscl 14677 | . . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) | |
36 | 12, 35 | sylan 583 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) |
37 | 34, 36 | elind 4121 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ)) |
38 | 37 | ne0d 4251 | . 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (𝐼 ∩ ℕ) ≠ ∅) |
39 | zringlpirlem.n0 | . . 3 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
40 | zring0 20173 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
41 | 9, 40 | lidlnz 19994 | . . 3 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
42 | 22, 7, 39, 41 | mp3an2i 1463 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
43 | 38, 42 | r19.29a 3248 | 1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ‘cfv 6324 ℂcc 10524 0cc0 10526 -cneg 10860 ℕcn 11625 ℤcz 11969 abscabs 14585 invgcminusg 18096 SubGrpcsubg 18265 Ringcrg 19290 SubRingcsubrg 19524 LIdealclidl 19935 ℂfldccnfld 20091 ℤringzring 20163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-cnfld 20092 df-zring 20164 |
This theorem is referenced by: zringlpirlem2 20178 zringlpirlem3 20179 |
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