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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzinico2 | Structured version Visualization version GIF version |
Description: An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
uzinico2.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
Ref | Expression |
---|---|
uzinico2 | ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4235 | . . . 4 ⊢ (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞)) = ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞)) = ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞)))) |
3 | incom 4216 | . . . . 5 ⊢ ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀)) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀))) |
5 | uzssz 12896 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘𝑀) ⊆ ℤ) |
7 | uzinico2.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
8 | 6, 7 | sseldd 3995 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
9 | eqid 2734 | . . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
10 | 8, 9 | uzinico 45512 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) = (ℤ ∩ (𝑁[,)+∞))) |
11 | 10 | eqcomd 2740 | . . . . 5 ⊢ (𝜑 → (ℤ ∩ (𝑁[,)+∞)) = (ℤ≥‘𝑁)) |
12 | 11 | ineq1d 4226 | . . . 4 ⊢ (𝜑 → ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀)) = ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀))) |
13 | 7 | uzssd 45357 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
14 | dfss2 3980 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀) ↔ ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) | |
15 | 13, 14 | sylib 218 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) |
16 | 4, 12, 15 | 3eqtrd 2778 | . . 3 ⊢ (𝜑 → ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = (ℤ≥‘𝑁)) |
17 | uzssz 12896 | . . . . . 6 ⊢ (ℤ≥‘𝑁) ⊆ ℤ | |
18 | dfss2 3980 | . . . . . 6 ⊢ ((ℤ≥‘𝑁) ⊆ ℤ ↔ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) | |
19 | 17, 18 | mpbi 230 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁) |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) |
21 | 20 | eqcomd 2740 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑁) ∩ ℤ)) |
22 | 2, 16, 21 | 3eqtrrd 2779 | . 2 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞))) |
23 | dfss2 3980 | . . . . 5 ⊢ ((ℤ≥‘𝑀) ⊆ ℤ ↔ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀)) | |
24 | 5, 23 | mpbi 230 | . . . 4 ⊢ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀) |
25 | 24 | ineq1i 4223 | . . 3 ⊢ (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞)) |
26 | 25 | a1i 11 | . 2 ⊢ (𝜑 → (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) |
27 | 22, 20, 26 | 3eqtr3d 2782 | 1 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∩ cin 3961 ⊆ wss 3962 ‘cfv 6562 (class class class)co 7430 +∞cpnf 11289 ℤcz 12610 ℤ≥cuz 12875 [,)cico 13385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-neg 11492 df-z 12611 df-uz 12876 df-ico 13389 |
This theorem is referenced by: uzinico3 45515 limsupvaluz 45663 |
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