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Mirrors > Home > MPE Home > Th. List > zringlpirlem1 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 20039. 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 809 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → 𝑎 ∈ 𝐼) | |
2 | eleq1 2827 | . . . . . 6 ⊢ ((abs‘𝑎) = 𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ 𝑎 ∈ 𝐼)) | |
3 | 1, 2 | syl5ibrcom 237 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 → (abs‘𝑎) ∈ 𝐼)) |
4 | zsubrg 20001 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
5 | subrgsubg 18988 | . . . . . . . . . . 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 20026 | . . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤring) | |
9 | eqid 2760 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
10 | 8, 9 | lidlss 19412 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ) |
12 | 11 | sselda 3744 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ) |
13 | df-zring 20021 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
14 | eqid 2760 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
15 | eqid 2760 | . . . . . . . . . . 11 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
16 | 13, 14, 15 | subginv 17802 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑎 ∈ ℤ) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
17 | 6, 12, 16 | sylancr 698 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
18 | 12 | zcnd 11675 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ) |
19 | cnfldneg 19974 | . . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((invg‘ℂfld)‘𝑎) = -𝑎) | |
20 | 18, 19 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = -𝑎) |
21 | 17, 20 | eqtr3d 2796 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) = -𝑎) |
22 | zringring 20023 | . . . . . . . . . 10 ⊢ ℤring ∈ Ring | |
23 | 22 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ℤring ∈ Ring) |
24 | 7 | adantr 472 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘ℤring)) |
25 | simpr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼) | |
26 | 9, 15 | lidlnegcl 19416 | . . . . . . . . 9 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
27 | 23, 24, 25, 26 | syl3anc 1477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
28 | 21, 27 | eqeltrrd 2840 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → -𝑎 ∈ 𝐼) |
29 | 28 | adantr 472 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → -𝑎 ∈ 𝐼) |
30 | eleq1 2827 | . . . . . 6 ⊢ ((abs‘𝑎) = -𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ -𝑎 ∈ 𝐼)) | |
31 | 29, 30 | syl5ibrcom 237 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = -𝑎 → (abs‘𝑎) ∈ 𝐼)) |
32 | 12 | zred 11674 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ) |
33 | 32 | absord 14353 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
34 | 33 | adantr 472 | . . . . 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 14264 | . . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) | |
37 | 12, 36 | sylan 489 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) |
38 | 35, 37 | elind 3941 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ)) |
39 | ne0i 4064 | . . 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 20030 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
44 | 9, 43 | lidlnz 19430 | . . 3 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
45 | 41, 7, 42, 44 | syl3anc 1477 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
46 | 40, 45 | r19.29a 3216 | 1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∃wrex 3051 ∩ cin 3714 ⊆ wss 3715 ∅c0 4058 {csn 4321 ‘cfv 6049 ℂcc 10126 0cc0 10128 -cneg 10459 ℕcn 11212 ℤcz 11569 abscabs 14173 invgcminusg 17624 SubGrpcsubg 17789 Ringcrg 18747 SubRingcsubrg 18978 LIdealclidl 19372 ℂfldccnfld 19948 ℤringzring 20020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-rp 12026 df-fz 12520 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-0g 16304 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-cmn 18395 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-subrg 18980 df-lmod 19067 df-lss 19135 df-sra 19374 df-rgmod 19375 df-lidl 19376 df-cnfld 19949 df-zring 20021 |
This theorem is referenced by: zringlpirlem2 20035 zringlpirlem3 20036 |
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