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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzinico3 | Structured version Visualization version GIF version |
Description: An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
uzinico3.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzinico3.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzinico3 | ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzinico3.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | 1 | uzidd 12247 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | 2 | uzinico2 41714 | . 2 ⊢ (𝜑 → (ℤ≥‘𝑀) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
4 | uzinico3.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 = (ℤ≥‘𝑀)) |
6 | 5 | ineq1d 4185 | . . 3 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
7 | 5, 6 | eqeq12d 2834 | . 2 ⊢ (𝜑 → (𝑍 = (𝑍 ∩ (𝑀[,)+∞)) ↔ (ℤ≥‘𝑀) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞)))) |
8 | 3, 7 | mpbird 258 | 1 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ‘cfv 6348 (class class class)co 7145 +∞cpnf 10660 ℤcz 11969 ℤ≥cuz 12231 [,)cico 12728 |
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-pre-lttri 10599 ax-pre-lttrn 10600 |
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-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 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-neg 10861 df-z 11970 df-uz 12232 df-ico 12732 |
This theorem is referenced by: liminfvaluz 41949 limsupvaluz3 41955 |
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