![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > zringlpirlem1 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 21495. 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 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → 𝑎 ∈ 𝐼) | |
2 | eleq1 2826 | . . . . . 6 ⊢ ((abs‘𝑎) = 𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ 𝑎 ∈ 𝐼)) | |
3 | 1, 2 | syl5ibrcom 247 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 → (abs‘𝑎) ∈ 𝐼)) |
4 | zsubrg 21455 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
5 | subrgsubg 20593 | . . . . . . . . . . 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 21481 | . . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤring) | |
9 | eqid 2734 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
10 | 8, 9 | lidlss 21239 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ) |
12 | 11 | sselda 3994 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ) |
13 | df-zring 21475 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
14 | eqid 2734 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
15 | eqid 2734 | . . . . . . . . . . 11 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
16 | 13, 14, 15 | subginv 19163 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑎 ∈ ℤ) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
17 | 6, 12, 16 | sylancr 587 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = ((invg‘ℤring)‘𝑎)) |
18 | 12 | zcnd 12720 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ) |
19 | cnfldneg 21425 | . . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((invg‘ℂfld)‘𝑎) = -𝑎) | |
20 | 18, 19 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = -𝑎) |
21 | 17, 20 | eqtr3d 2776 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) = -𝑎) |
22 | zringring 21477 | . . . . . . . . 9 ⊢ ℤring ∈ Ring | |
23 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘ℤring)) |
24 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼) | |
25 | 9, 15 | lidlnegcl 21249 | . . . . . . . . 9 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
26 | 22, 23, 24, 25 | mp3an2i 1465 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
27 | 21, 26 | eqeltrrd 2839 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → -𝑎 ∈ 𝐼) |
28 | 27 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → -𝑎 ∈ 𝐼) |
29 | eleq1 2826 | . . . . . 6 ⊢ ((abs‘𝑎) = -𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ -𝑎 ∈ 𝐼)) | |
30 | 28, 29 | syl5ibrcom 247 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = -𝑎 → (abs‘𝑎) ∈ 𝐼)) |
31 | 12 | zred 12719 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ) |
32 | 31 | absord 15450 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
33 | 32 | adantr 480 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 ∨ (abs‘𝑎) = -𝑎)) |
34 | 3, 30, 33 | mpjaod 860 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ 𝐼) |
35 | nnabscl 15360 | . . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) | |
36 | 12, 35 | sylan 580 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ) |
37 | 34, 36 | elind 4209 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ)) |
38 | 37 | ne0d 4347 | . 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (𝐼 ∩ ℕ) ≠ ∅) |
39 | zringlpirlem.n0 | . . 3 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
40 | zring0 21486 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
41 | 9, 40 | lidlnz 21269 | . . 3 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
42 | 22, 7, 39, 41 | mp3an2i 1465 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
43 | 38, 42 | r19.29a 3159 | 1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 {csn 4630 ‘cfv 6562 ℂcc 11150 0cc0 11152 -cneg 11490 ℕcn 12263 ℤcz 12610 abscabs 15269 invgcminusg 18964 SubGrpcsubg 19150 Ringcrg 20250 SubRingcsubrg 20585 LIdealclidl 21233 ℂfldccnfld 21381 ℤringczring 21474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-lmod 20876 df-lss 20947 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-cnfld 21382 df-zring 21475 |
This theorem is referenced by: zringlpirlem2 21491 zringlpirlem3 21492 |
Copyright terms: Public domain | W3C validator |