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Mirrors > Home > MPE Home > Th. List > fzoun | Structured version Visualization version GIF version |
Description: A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.) |
Ref | Expression |
---|---|
fzoun | ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12236 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℤ) |
3 | eluzelz 12241 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
4 | nn0z 11993 | . . . . 5 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ) | |
5 | zaddcl 12010 | . . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 + 𝐶) ∈ ℤ) | |
6 | 3, 4, 5 | syl2an 595 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐵 + 𝐶) ∈ ℤ) |
7 | 3 | adantr 481 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ ℤ) |
8 | 2, 6, 7 | 3jca 1120 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴 ∈ ℤ ∧ (𝐵 + 𝐶) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
9 | eluzle 12244 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ≤ 𝐵) | |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 𝐴 ≤ 𝐵) |
11 | nn0ge0 11910 | . . . . . 6 ⊢ (𝐶 ∈ ℕ0 → 0 ≤ 𝐶) | |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 0 ≤ 𝐶) |
13 | eluzelre 12242 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℝ) | |
14 | nn0re 11894 | . . . . . 6 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℝ) | |
15 | addge01 11138 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (0 ≤ 𝐶 ↔ 𝐵 ≤ (𝐵 + 𝐶))) | |
16 | 13, 14, 15 | syl2an 595 | . . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (0 ≤ 𝐶 ↔ 𝐵 ≤ (𝐵 + 𝐶))) |
17 | 12, 16 | mpbid 233 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 𝐵 ≤ (𝐵 + 𝐶)) |
18 | 10, 17 | jca 512 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ (𝐵 + 𝐶))) |
19 | elfz2 12887 | . . 3 ⊢ (𝐵 ∈ (𝐴...(𝐵 + 𝐶)) ↔ ((𝐴 ∈ ℤ ∧ (𝐵 + 𝐶) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ (𝐵 + 𝐶)))) | |
20 | 8, 18, 19 | sylanbrc 583 | . 2 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ (𝐴...(𝐵 + 𝐶))) |
21 | fzosplit 13058 | . 2 ⊢ (𝐵 ∈ (𝐴...(𝐵 + 𝐶)) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶)))) | |
22 | 20, 21 | syl 17 | 1 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 + caddc 10528 ≤ cle 10664 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 ..^cfzo 13021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 |
This theorem is referenced by: clwwlkccatlem 27694 |
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