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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 41143 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | 2 | uzinico2 41303 | . 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 4069 | . . 3 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
7 | 5, 6 | eqeq12d 2786 | . 2 ⊢ (𝜑 → (𝑍 = (𝑍 ∩ (𝑀[,)+∞)) ↔ (ℤ≥‘𝑀) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞)))) |
8 | 3, 7 | mpbird 249 | 1 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ∩ cin 3821 ‘cfv 6185 (class class class)co 6974 +∞cpnf 10469 ℤcz 11791 ℤ≥cuz 12056 [,)cico 12554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-pre-lttri 10407 ax-pre-lttrn 10408 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-neg 10671 df-z 11792 df-uz 12057 df-ico 12558 |
This theorem is referenced by: liminfvaluz 41538 limsupvaluz3 41544 |
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