![]() |
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fzsplitnd | Structured version Visualization version GIF version |
Description: Split a finite interval of integers into two parts. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
fzsplitnd.1 | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
Ref | Expression |
---|---|
fzsplitnd | ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzsplitnd.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
2 | elfzuz 12937 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
4 | 1 | elfzelzd 12942 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | 4 | zcnd 12112 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
6 | 1cnd 10659 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 11024 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
8 | 7 | eleq1d 2835 | . . . 4 ⊢ (𝜑 → (((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ↔ 𝐾 ∈ (ℤ≥‘𝑀))) |
9 | 3, 8 | mpbird 260 | . . 3 ⊢ (𝜑 → ((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀)) |
10 | 1zzd 12037 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | 4, 10 | zsubcld 12116 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
12 | elfzuz3 12938 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
14 | 7 | fveq2d 6655 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
15 | 14 | eleq2d 2836 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝐾))) |
16 | 13, 15 | mpbird 260 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
17 | peano2uzr 12328 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) | |
18 | 11, 16, 17 | syl2anc 588 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
19 | fzsplit2 12966 | . . 3 ⊢ ((((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) | |
20 | 9, 18, 19 | syl2anc 588 | . 2 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) |
21 | 7 | oveq1d 7158 | . . 3 ⊢ (𝜑 → (((𝐾 − 1) + 1)...𝑁) = (𝐾...𝑁)) |
22 | 21 | uneq2d 4064 | . 2 ⊢ (𝜑 → ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁)) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
23 | 20, 22 | eqtrd 2794 | 1 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ∪ cun 3852 ‘cfv 6328 (class class class)co 7143 1c1 10561 + caddc 10563 − cmin 10893 ℤcz 12005 ℤ≥cuz 12267 ...cfz 12924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 |
This theorem is referenced by: fzsplitnr 39536 metakunt24 39655 |
Copyright terms: Public domain | W3C validator |