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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 12754 | . . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | peano2uz 12150 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) |
4 | elfzuz3 12755 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
5 | fzsplit2 12782 | . 2 ⊢ (((𝐾 + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) | |
6 | 3, 4, 5 | syl2anc 584 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∪ cun 3857 ‘cfv 6225 (class class class)co 7016 1c1 10384 + caddc 10386 ℤ≥cuz 12093 ...cfz 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 |
This theorem is referenced by: fzsuc 12804 fz0to3un2pr 12859 fz0sn0fz1 12874 fsum1p 14941 o1fsum 15001 climcndslem1 15037 climcndslem2 15038 mertenslem1 15073 fprod1p 15155 fprodeq0 15162 4sqlem19 16128 uniioombllem3 23869 mtest 24675 birthdaylem2 25212 ftalem5 25336 chtdif 25417 ppidif 25422 gausslemma2dlem4 25627 lgsquadlem2 25639 pntpbnd2 25845 axlowdimlem3 26413 axlowdimlem16 26426 axlowdimlem17 26427 esumpmono 30955 fsum2dsub 31495 subfacp1lem1 32034 subfacp1lem5 32039 poimirlem1 34424 poimirlem2 34425 poimirlem6 34429 poimirlem7 34430 poimirlem8 34431 poimirlem11 34434 fzsplit1nn0 38836 stoweidlem11 41838 stoweidlem26 41853 31prm 43242 |
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