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Mirrors > Home > MPE Home > Th. List > uztrn | Structured version Visualization version GIF version |
Description: Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
uztrn | ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 11568 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℤ) |
3 | eluzelz 11573 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
5 | eluzle 11576 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝐾) |
7 | eluzle 11576 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝐾 ≤ 𝑀) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ≤ 𝑀) |
9 | eluzelz 11573 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℤ) |
11 | zletr 11298 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) | |
12 | 2, 10, 4, 11 | syl3anc 1318 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) |
13 | 6, 8, 12 | mp2and 711 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑀) |
14 | eluz2 11569 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀)) | |
15 | 2, 4, 13, 14 | syl3anbrc 1239 | 1 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 ≤ cle 9954 ℤcz 11254 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 |
This theorem is referenced by: uztrn2 11581 fzsplit2 12237 fzass4 12250 fzss1 12251 fzss2 12252 uzsplit 12281 seqfveq2 12685 sermono 12695 seqsplit 12696 seqid2 12709 fzsdom2 13075 seqcoll 13105 spllen 13356 splfv2a 13358 splval2 13359 climcndslem1 14420 mertenslem1 14455 ntrivcvgfvn0 14470 zprod 14506 dvdsfac 14886 smupvallem 15043 vdwlem2 15524 vdwlem6 15528 efgredleme 17979 bposlem6 24814 dchrisumlem2 24979 axlowdimlem16 25637 fzsplit3 28940 sseqf 29781 ballotlemsima 29904 ballotlemfrc 29915 climuzcnv 30819 seqpo 32713 incsequz2 32715 mettrifi 32723 monotuz 36524 |
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