![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fzsplit | Structured version Visualization version GIF version |
Description: Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
fzsplit | ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 12376 | . . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 11779 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) |
4 | elfzuz3 12377 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
5 | fzsplit2 12404 | . 2 ⊢ (((𝐾 + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) | |
6 | 3, 4, 5 | syl2anc 694 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 ‘cfv 5926 (class class class)co 6690 1c1 9975 + caddc 9977 ℤ≥cuz 11725 ...cfz 12364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 |
This theorem is referenced by: fzsuc 12426 fz0to3un2pr 12480 fz0sn0fz1 12495 fsum1p 14526 o1fsum 14589 climcndslem1 14625 climcndslem2 14626 mertenslem1 14660 fprod1p 14742 fprodeq0 14749 4sqlem19 15714 uniioombllem3 23399 iblcnlem1 23599 mtest 24203 birthdaylem2 24724 ftalem5 24848 chtdif 24929 ppidif 24934 gausslemma2dlem4 25139 lgsquadlem2 25151 pntpbnd2 25321 axlowdimlem3 25869 axlowdimlem16 25882 axlowdimlem17 25883 esumpmono 30269 fsum2dsub 30813 subfacp1lem1 31287 subfacp1lem5 31292 poimirlem1 33540 poimirlem2 33541 poimirlem6 33545 poimirlem7 33546 poimirlem8 33547 poimirlem11 33550 fzsplit1nn0 37634 stoweidlem11 40546 stoweidlem26 40561 31prm 41837 |
Copyright terms: Public domain | W3C validator |