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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 13444 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
4 | 1 | elfzelzd 13449 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | 4 | zcnd 12615 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
6 | 1cnd 11157 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 11523 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
8 | 7 | eleq1d 2823 | . . . 4 ⊢ (𝜑 → (((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ↔ 𝐾 ∈ (ℤ≥‘𝑀))) |
9 | 3, 8 | mpbird 257 | . . 3 ⊢ (𝜑 → ((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀)) |
10 | 1zzd 12541 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | 4, 10 | zsubcld 12619 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
12 | elfzuz3 13445 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
14 | 7 | fveq2d 6851 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
15 | 14 | eleq2d 2824 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝐾))) |
16 | 13, 15 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
17 | peano2uzr 12835 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) | |
18 | 11, 16, 17 | syl2anc 585 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
19 | fzsplit2 13473 | . . 3 ⊢ ((((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) | |
20 | 9, 18, 19 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) |
21 | 7 | oveq1d 7377 | . . 3 ⊢ (𝜑 → (((𝐾 − 1) + 1)...𝑁) = (𝐾...𝑁)) |
22 | 21 | uneq2d 4128 | . 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 1542 ∈ wcel 2107 ∪ cun 3913 ‘cfv 6501 (class class class)co 7362 1c1 11059 + caddc 11061 − cmin 11392 ℤcz 12506 ℤ≥cuz 12770 ...cfz 13431 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 |
This theorem is referenced by: fzsplitnr 40470 metakunt24 40629 |
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