zringlpirlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > zringlpirlem1		Structured version Visualization version GIF version

Theorem zringlpirlem1 20596

Description: Lemma for zringlpir 20601. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)

**Hypotheses**
Ref	Expression
zringlpirlem.i	⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤ_ring))
zringlpirlem.n0	⊢ (𝜑 → 𝐼 ≠ {0})

**Assertion**
Ref	Expression
zringlpirlem1	⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅)

Proof of Theorem zringlpirlem1 Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → 𝑎 ∈ 𝐼)
2		eleq1 2826	. . . . . 6 ⊢ ((abs‘𝑎) = 𝑎 → ((abs‘𝑎) ∈ 𝐼 ↔ 𝑎 ∈ 𝐼))
3	1, 2	syl5ibrcom 246	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = 𝑎 → (abs‘𝑎) ∈ 𝐼))
4		zsubrg 20563	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
5		subrgsubg 19945	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
7		zringlpirlem.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤ_ring))
8		zringbas 20588	. . . . . . . . . . . . 13 ⊢ ℤ = (Base‘ℤ_ring)
9		eqid 2738	. . . . . . . . . . . . 13 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
10	8, 9	lidlss 20394	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘ℤ_ring) → 𝐼 ⊆ ℤ)
11	7, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ℤ)
12	11	sselda 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℤ)
13		df-zring 20583	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
14		eqid 2738	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
15		eqid 2738	. . . . . . . . . . 11 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
16	13, 14, 15	subginv 18677	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝑎 ∈ ℤ) → ((inv_g‘ℂ_fld)‘𝑎) = ((inv_g‘ℤ_ring)‘𝑎))
17	6, 12, 16	sylancr 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((inv_g‘ℂ_fld)‘𝑎) = ((inv_g‘ℤ_ring)‘𝑎))
18	12	zcnd 12356	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℂ)
19		cnfldneg 20536	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℂ → ((inv_g‘ℂ_fld)‘𝑎) = -𝑎)
20	18, 19	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((inv_g‘ℂ_fld)‘𝑎) = -𝑎)
21	17, 20	eqtr3d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((inv_g‘ℤ_ring)‘𝑎) = -𝑎)
22		zringring 20585	. . . . . . . . 9 ⊢ ℤ_ring ∈ Ring
23	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ (LIdeal‘ℤ_ring))
24		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
25	9, 15	lidlnegcl 20398	. . . . . . . . 9 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤ_ring) ∧ 𝑎 ∈ 𝐼) → ((inv_g‘ℤ_ring)‘𝑎) ∈ 𝐼)
26	22, 23, 24, 25	mp3an2i 1464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((inv_g‘ℤ_ring)‘𝑎) ∈ 𝐼)
27	21, 26	eqeltrrd 2840	. . . . . . 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 246	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → ((abs‘𝑎) = -𝑎 → (abs‘𝑎) ∈ 𝐼))
31	12	zred 12355	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ ℝ)
32	31	absord 15055	. . . . . 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 856	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ 𝐼)
35		nnabscl 14965	. . . . 5 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
36	12, 35	sylan 579	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
37	34, 36	elind 4124	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ (𝐼 ∩ ℕ))
38	37	ne0d 4266	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐼) ∧ 𝑎 ≠ 0) → (𝐼 ∩ ℕ) ≠ ∅)
39		zringlpirlem.n0	. . 3 ⊢ (𝜑 → 𝐼 ≠ {0})
40		zring0 20592	. . . 4 ⊢ 0 = (0_g‘ℤ_ring)
41	9, 40	lidlnz 20412	. . 3 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐼 ∈ (LIdeal‘ℤ_ring) ∧ 𝐼 ≠ {0}) → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0)
42	22, 7, 39, 41	mp3an2i 1464	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐼 𝑎 ≠ 0)
43	38, 42	r19.29a 3217	1 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 ‘cfv 6418 ℂcc 10800 0cc0 10802 -cneg 11136 ℕcn 11903 ℤcz 12249 abscabs 14873 inv_gcminusg 18493 SubGrpcsubg 18664 Ringcrg 19698 SubRingcsubrg 19935 LIdealclidl 20347 ℂ_fldccnfld 20510 ℤ_ringzring 20582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-cnfld 20511 df-zring 20583

This theorem is referenced by: zringlpirlem2 20597 zringlpirlem3 20598

W3C validator