Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > zringlpirlem2 | Structured version Visualization version GIF version |
Description: Lemma for zringlpir 20638. 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 4208 | . . . . 5 ⊢ (𝐼 ∩ ℕ) ⊆ ℕ | |
3 | nnuz 12284 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 4005 | . . . 4 ⊢ (𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) |
5 | zringlpirlem.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) | |
6 | zringlpirlem.n0 | . . . . 5 ⊢ (𝜑 → 𝐼 ≠ {0}) | |
7 | 5, 6 | zringlpirlem1 20633 | . . . 4 ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) |
8 | infssuzcl 12335 | . . . 4 ⊢ (((𝐼 ∩ ℕ) ⊆ (ℤ≥‘1) ∧ (𝐼 ∩ ℕ) ≠ ∅) → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) | |
9 | 4, 7, 8 | sylancr 589 | . . 3 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ (𝐼 ∩ ℕ)) |
10 | 9 | elin1d 4177 | . 2 ⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈ 𝐼) |
11 | 1, 10 | eqeltrid 2919 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 ‘cfv 6357 infcinf 8907 ℝcr 10538 0cc0 10539 1c1 10540 < clt 10677 ℕcn 11640 ℤ≥cuz 12246 LIdealclidl 19944 ℤringzring 20619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-cnfld 20548 df-zring 20620 |
This theorem is referenced by: zringlpirlem3 20635 zringlpir 20638 |
Copyright terms: Public domain | W3C validator |