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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzspl | Structured version Visualization version GIF version |
Description: Split the last element of a finite set of sequential integers. More generic than fzsuc 13545. (Contributed by Thierry Arnoux, 7-Nov-2016.) |
Ref | Expression |
---|---|
fzspl | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12829 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 12664 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) |
3 | 1zzd 12590 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 1 ∈ ℤ) | |
4 | 3 | zcnd 12664 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 1 ∈ ℂ) |
5 | 2, 4 | npcand 11572 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑁 − 1) + 1) = 𝑁) |
6 | 5 | eleq1d 2810 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (ℤ≥‘𝑀))) |
7 | 6 | ibir 268 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑀)) |
8 | eluzelre 12830 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
9 | 8 | lem1d 12144 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 1) ≤ 𝑁) |
10 | 1, 3 | zsubcld 12668 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 1) ∈ ℤ) |
11 | eluz1 12823 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) ↔ (𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁))) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) ↔ (𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁))) |
13 | 1, 9, 12 | mpbir2and 710 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
14 | fzsplit2 13523 | . . 3 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) | |
15 | 7, 13, 14 | syl2anc 583 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
16 | 5 | oveq1d 7416 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁)) |
17 | fzsn 13540 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) | |
18 | 1, 17 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁...𝑁) = {𝑁}) |
19 | 16, 18 | eqtrd 2764 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝑁 − 1) + 1)...𝑁) = {𝑁}) |
20 | 19 | uneq2d 4155 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
21 | 15, 20 | eqtrd 2764 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cun 3938 {csn 4620 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 1c1 11107 + caddc 11109 ≤ cle 11246 − cmin 11441 ℤcz 12555 ℤ≥cuz 12819 ...cfz 13481 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 |
This theorem is referenced by: fzdif2 32471 ballotlemfp1 33979 |
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