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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 13529 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
4 | 1 | elfzelzd 13534 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | 4 | zcnd 12697 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
6 | 1cnd 11239 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
7 | 5, 6 | npcand 11605 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
8 | 7 | eleq1d 2810 | . . . 4 ⊢ (𝜑 → (((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ↔ 𝐾 ∈ (ℤ≥‘𝑀))) |
9 | 3, 8 | mpbird 256 | . . 3 ⊢ (𝜑 → ((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀)) |
10 | 1zzd 12623 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | 4, 10 | zsubcld 12701 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
12 | elfzuz3 13530 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) |
14 | 7 | fveq2d 6896 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘((𝐾 − 1) + 1)) = (ℤ≥‘𝐾)) |
15 | 14 | eleq2d 2811 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1)) ↔ 𝑁 ∈ (ℤ≥‘𝐾))) |
16 | 13, 15 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) |
17 | peano2uzr 12917 | . . . 4 ⊢ (((𝐾 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝐾 − 1) + 1))) → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) | |
18 | 11, 16, 17 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) |
19 | fzsplit2 13558 | . . 3 ⊢ ((((𝐾 − 1) + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐾 − 1))) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) | |
20 | 9, 18, 19 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁))) |
21 | 7 | oveq1d 7431 | . . 3 ⊢ (𝜑 → (((𝐾 − 1) + 1)...𝑁) = (𝐾...𝑁)) |
22 | 21 | uneq2d 4156 | . 2 ⊢ (𝜑 → ((𝑀...(𝐾 − 1)) ∪ (((𝐾 − 1) + 1)...𝑁)) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
23 | 20, 22 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3937 ‘cfv 6543 (class class class)co 7416 1c1 11139 + caddc 11141 − cmin 11474 ℤcz 12588 ℤ≥cuz 12852 ...cfz 13516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 |
This theorem is referenced by: fzsplitnr 41510 metakunt24 41736 |
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