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 13108 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
4 | 1 | elfzelzd 13113 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | 4 | zcnd 12283 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
6 | 1cnd 10828 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 11193 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
8 | 7 | eleq1d 2822 | . . . 4 ⊢ (𝜑 → (((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ↔ 𝐾 ∈ (ℤ≥‘𝑀))) |
9 | 3, 8 | mpbird 260 | . . 3 ⊢ (𝜑 → ((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀)) |
10 | 1zzd 12208 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | 4, 10 | zsubcld 12287 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
12 | elfzuz3 13109 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
14 | 7 | fveq2d 6721 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
15 | 14 | eleq2d 2823 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝐾))) |
16 | 13, 15 | mpbird 260 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
17 | peano2uzr 12499 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) | |
18 | 11, 16, 17 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
19 | fzsplit2 13137 | . . 3 ⊢ ((((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) | |
20 | 9, 18, 19 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) |
21 | 7 | oveq1d 7228 | . . 3 ⊢ (𝜑 → (((𝐾 − 1) + 1)...𝑁) = (𝐾...𝑁)) |
22 | 21 | uneq2d 4077 | . 2 ⊢ (𝜑 → ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁)) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
23 | 20, 22 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 ‘cfv 6380 (class class class)co 7213 1c1 10730 + caddc 10732 − cmin 11062 ℤcz 12176 ℤ≥cuz 12438 ...cfz 13095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 |
This theorem is referenced by: fzsplitnr 39726 metakunt24 39870 |
Copyright terms: Public domain | W3C validator |