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Mathbox for Glauco Siliprandi |
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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 12842 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | 2 | uzinico2 44844 | . 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 4206 | . . 3 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
7 | 5, 6 | eqeq12d 2742 | . 2 ⊢ (𝜑 → (𝑍 = (𝑍 ∩ (𝑀[,)+∞)) ↔ (ℤ≥‘𝑀) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞)))) |
8 | 3, 7 | mpbird 257 | 1 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ‘cfv 6537 (class class class)co 7405 +∞cpnf 11249 ℤcz 12562 ℤ≥cuz 12826 [,)cico 13332 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-ico 13336 |
This theorem is referenced by: liminfvaluz 45077 limsupvaluz3 45083 |
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