![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > zringlpirlem2 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 20156. 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 | inss1 4026 | . . 3 ⊢ (𝐼 ∩ ℕ) ⊆ 𝐼 | |
3 | inss2 4027 | . . . . 5 ⊢ (𝐼 ∩ ℕ) ⊆ ℕ | |
4 | nnuz 11963 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
5 | 3, 4 | sseqtri 3831 | . . . 4 ⊢ (𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) |
6 | zringlpirlem.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) | |
7 | zringlpirlem.n0 | . . . . 5 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
8 | 6, 7 | zringlpirlem1 20151 | . . . 4 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
9 | infssuzcl 12013 | . . . 4 ⊢ (((𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) ∧ (𝐼 ∩ ℕ) ≠ ∅) → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) | |
10 | 5, 8, 9 | sylancr 582 | . . 3 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) |
11 | 2, 10 | sseldi 3794 | . 2 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ 𝐼) |
12 | 1, 11 | syl5eqel 2880 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∩ cin 3766 ⊆ wss 3767 ∅c0 4113 {csn 4366 ‘cfv 6099 infcinf 8587 ℝcr 10221 0cc0 10222 1c1 10223 < clt 10361 ℕcn 11310 ℤ≥cuz 11926 LIdealclidl 19490 ℤringzring 20137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-rp 12071 df-fz 12577 df-seq 13052 df-exp 13111 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-sbg 17740 df-subg 17901 df-cmn 18507 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-subrg 19093 df-lmod 19180 df-lss 19248 df-sra 19492 df-rgmod 19493 df-lidl 19494 df-cnfld 20066 df-zring 20138 |
This theorem is referenced by: zringlpirlem3 20153 zringlpir 20156 |
Copyright terms: Public domain | W3C validator |