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Mirrors > Home > MPE Home > Th. List > zringlpirlem2 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 21453. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Revised by AV, 27-Sep-2020.) |
Ref | Expression |
---|---|
zringlpirlem.i | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) |
zringlpirlem.n0 | ⊢ (𝜑 → 𝐼 ≠ {0}) |
zringlpirlem.g | ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < ) |
Ref | Expression |
---|---|
zringlpirlem2 | ⊢ (𝜑 → 𝐺 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringlpirlem.g | . 2 ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < ) | |
2 | inss2 4228 | . . . . 5 ⊢ (𝐼 ∩ ℕ) ⊆ ℕ | |
3 | nnuz 12911 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 4015 | . . . 4 ⊢ (𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) |
5 | zringlpirlem.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) | |
6 | zringlpirlem.n0 | . . . . 5 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
7 | 5, 6 | zringlpirlem1 21448 | . . . 4 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
8 | infssuzcl 12962 | . . . 4 ⊢ (((𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) ∧ (𝐼 ∩ ℕ) ≠ ∅) → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) | |
9 | 4, 7, 8 | sylancr 585 | . . 3 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) |
10 | 9 | elin1d 4196 | . 2 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ 𝐼) |
11 | 1, 10 | eqeltrid 2830 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∩ cin 3945 ⊆ wss 3946 ∅c0 4322 {csn 4623 ‘cfv 6546 infcinf 9477 ℝcr 11148 0cc0 11149 1c1 11150 < clt 11289 ℕcn 12258 ℤ≥cuz 12868 LIdealclidl 21191 ℤringczring 21432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-subrng 20524 df-subrg 20549 df-lmod 20834 df-lss 20905 df-sra 21147 df-rgmod 21148 df-lidl 21193 df-cnfld 21340 df-zring 21433 |
This theorem is referenced by: zringlpirlem3 21450 zringlpir 21453 |
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