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| Mirrors > Home > MPE Home > Th. List > zringlpirlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for zringlpir 21478. 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 2829 | . . . . . 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 21438 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
| 5 | subrgsubg 20577 | . . . . . . . . . . 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 21464 | . . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤring) | |
| 9 | eqid 2737 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤring) = (LIdeal‘ℤring) | |
| 10 | 8, 9 | lidlss 21222 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
| 11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ) |
| 12 | 11 | sselda 3983 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ) |
| 13 | df-zring 21458 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
| 14 | eqid 2737 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 15 | eqid 2737 | . . . . . . . . . . 11 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
| 16 | 13, 14, 15 | subginv 19151 | . . . . . . . . . 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 12723 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ) |
| 19 | cnfldneg 21408 | . . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((invg‘ℂfld)‘𝑎) = -𝑎) | |
| 20 | 18, 19 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℂfld)‘𝑎) = -𝑎) |
| 21 | 17, 20 | eqtr3d 2779 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) = -𝑎) |
| 22 | zringring 21460 | . . . . . . . . 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 21232 | . . . . . . . . 9 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
| 26 | 22, 23, 24, 25 | mp3an2i 1468 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((invg‘ℤring)‘𝑎) ∈ 𝐼) |
| 27 | 21, 26 | eqeltrrd 2842 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → -𝑎 ∈ 𝐼) |
| 28 | 27 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → -𝑎 ∈ 𝐼) |
| 29 | eleq1 2829 | . . . . . 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 12722 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ) |
| 32 | 31 | absord 15454 | . . . . . 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 861 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ 𝐼) |
| 35 | nnabscl 15364 | . . . . 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 4200 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ)) |
| 38 | 37 | ne0d 4342 | . 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (𝐼 ∩ ℕ) ≠ ∅) |
| 39 | zringlpirlem.n0 | . . 3 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
| 40 | zring0 21469 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
| 41 | 9, 40 | lidlnz 21252 | . . 3 ⊢ ((ℤring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
| 42 | 22, 7, 39, 41 | mp3an2i 1468 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0) |
| 43 | 38, 42 | r19.29a 3162 | 1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 ‘cfv 6561 ℂcc 11153 0cc0 11155 -cneg 11493 ℕcn 12266 ℤcz 12613 abscabs 15273 invgcminusg 18952 SubGrpcsubg 19138 Ringcrg 20230 SubRingcsubrg 20569 LIdealclidl 21216 ℂfldccnfld 21364 ℤringczring 21457 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-cnfld 21365 df-zring 21458 |
| This theorem is referenced by: zringlpirlem2 21474 zringlpirlem3 21475 |
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