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 4196 | . . . 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 4178 | . . . . 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 12265 | . . . . . . . . 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 3968 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
9 | eqid 2821 | . . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
10 | 8, 9 | uzinico 41856 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) = (ℤ ∩ (𝑁[,)+∞))) |
11 | 10 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → (ℤ ∩ (𝑁[,)+∞)) = (ℤ≥‘𝑁)) |
12 | 11 | ineq1d 4188 | . . . 4 ⊢ (𝜑 → ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀)) = ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀))) |
13 | 7 | uzssd 41701 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
14 | df-ss 3952 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀) ↔ ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) | |
15 | 13, 14 | sylib 220 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) |
16 | 4, 12, 15 | 3eqtrd 2860 | . . 3 ⊢ (𝜑 → ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = (ℤ≥‘𝑁)) |
17 | uzssz 12265 | . . . . . 6 ⊢ (ℤ≥‘𝑁) ⊆ ℤ | |
18 | df-ss 3952 | . . . . . 6 ⊢ ((ℤ≥‘𝑁) ⊆ ℤ ↔ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) | |
19 | 17, 18 | mpbi 232 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁) |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) |
21 | 20 | eqcomd 2827 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑁) ∩ ℤ)) |
22 | 2, 16, 21 | 3eqtrrd 2861 | . 2 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞))) |
23 | df-ss 3952 | . . . . 5 ⊢ ((ℤ≥‘𝑀) ⊆ ℤ ↔ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀)) | |
24 | 5, 23 | mpbi 232 | . . . 4 ⊢ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀) |
25 | 24 | ineq1i 4185 | . . 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 2864 | 1 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 +∞cpnf 10672 ℤcz 11982 ℤ≥cuz 12244 [,)cico 12741 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-ico 12745 |
This theorem is referenced by: uzinico3 41859 limsupvaluz 42009 |
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