![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 12843 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | 2 | uzinico2 44574 | . 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 4211 | . . 3 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
7 | 5, 6 | eqeq12d 2747 | . 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 1540 ∈ wcel 2105 ∩ cin 3947 ‘cfv 6543 (class class class)co 7412 +∞cpnf 11250 ℤcz 12563 ℤ≥cuz 12827 [,)cico 13331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-pre-lttri 11188 ax-pre-lttrn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-neg 11452 df-z 12564 df-uz 12828 df-ico 13335 |
This theorem is referenced by: liminfvaluz 44807 limsupvaluz3 44813 |
Copyright terms: Public domain | W3C validator |