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Mirrors > Home > MPE Home > Th. List > zringlpirlem1 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 19656. 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 788 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → 𝑎 ∈ 𝐼) | |
2 | eleq1 2676 | . . . . . 6 ⊢ ((abs‘𝑎) = 𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ 𝑎 ∈ 𝐼)) | |
3 | 1, 2 | syl5ibrcom 236 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 → (abs‘𝑎) ∈ 𝐼)) |
4 | zsubrg 19618 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
5 | subrgsubg 18609 | . . . . . . . . . . 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 19643 | . . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤring) | |
9 | eqid 2610 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
10 | 8, 9 | lidlss 19031 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ) |
12 | 11 | sselda 3568 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ) |
13 | df-zring 19638 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
14 | eqid 2610 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
15 | eqid 2610 | . . . . . . . . . . 11 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
16 | 13, 14, 15 | subginv 17424 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑎 ∈ ℤ) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
17 | 6, 12, 16 | sylancr 694 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
18 | 12 | zcnd 11359 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ) |
19 | cnfldneg 19591 | . . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((invg‘ℂfld)‘𝑎) = -𝑎) | |
20 | 18, 19 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = -𝑎) |
21 | 17, 20 | eqtr3d 2646 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) = -𝑎) |
22 | zringring 19640 | . . . . . . . . . 10 ⊢ ℤring ∈ Ring | |
23 | 22 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ℤring ∈ Ring) |
24 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘ℤring)) |
25 | simpr 476 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼) | |
26 | 9, 15 | lidlnegcl 19035 | . . . . . . . . 9 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
27 | 23, 24, 25, 26 | syl3anc 1318 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
28 | 21, 27 | eqeltrrd 2689 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → -𝑎 ∈ 𝐼) |
29 | 28 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → -𝑎 ∈ 𝐼) |
30 | eleq1 2676 | . . . . . 6 ⊢ ((abs‘𝑎) = -𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ -𝑎 ∈ 𝐼)) | |
31 | 29, 30 | syl5ibrcom 236 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = -𝑎 → (abs‘𝑎) ∈ 𝐼)) |
32 | 12 | zred 11358 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ) |
33 | 32 | absord 14002 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
34 | 33 | adantr 480 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
35 | 3, 31, 34 | mpjaod 395 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ 𝐼) |
36 | nnabscl 13913 | . . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) | |
37 | 12, 36 | sylan 487 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) |
38 | 35, 37 | elind 3760 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ)) |
39 | ne0i 3880 | . . 3 ⊢ ((abs‘𝑎) ∈ (𝐼 ∩ ℕ) → (𝐼 ∩ ℕ) ≠ ∅) | |
40 | 38, 39 | syl 17 | . 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (𝐼 ∩ ℕ) ≠ ∅) |
41 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → ℤring ∈ Ring) |
42 | zringlpirlem.n0 | . . 3 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
43 | zring0 19647 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
44 | 9, 43 | lidlnz 19049 | . . 3 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
45 | 41, 7, 42, 44 | syl3anc 1318 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
46 | 40, 45 | r19.29a 3060 | 1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ‘cfv 5804 ℂcc 9813 0cc0 9815 -cneg 10146 ℕcn 10897 ℤcz 11254 abscabs 13822 invgcminusg 17246 SubGrpcsubg 17411 Ringcrg 18370 SubRingcsubrg 18599 LIdealclidl 18991 ℂfldccnfld 19567 ℤringzring 19637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-cnfld 19568 df-zring 19638 |
This theorem is referenced by: zringlpirlem2 19652 zringlpirlem3 19653 |
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