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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 4180 | . . . 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 4162 | . . . . 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 12785 | . . . . . . . . 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 3946 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
9 | eqid 2737 | . . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
10 | 8, 9 | uzinico 43805 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) = (ℤ ∩ (𝑁[,)+∞))) |
11 | 10 | eqcomd 2743 | . . . . 5 ⊢ (𝜑 → (ℤ ∩ (𝑁[,)+∞)) = (ℤ≥‘𝑁)) |
12 | 11 | ineq1d 4172 | . . . 4 ⊢ (𝜑 → ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀)) = ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀))) |
13 | 7 | uzssd 43650 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
14 | df-ss 3928 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀) ↔ ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) | |
15 | 13, 14 | sylib 217 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘𝑁)) |
16 | 4, 12, 15 | 3eqtrd 2781 | . . 3 ⊢ (𝜑 → ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = (ℤ≥‘𝑁)) |
17 | uzssz 12785 | . . . . . 6 ⊢ (ℤ≥‘𝑁) ⊆ ℤ | |
18 | df-ss 3928 | . . . . . 6 ⊢ ((ℤ≥‘𝑁) ⊆ ℤ ↔ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) | |
19 | 17, 18 | mpbi 229 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁) |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (ℤ≥‘𝑁)) |
21 | 20 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑁) ∩ ℤ)) |
22 | 2, 16, 21 | 3eqtrrd 2782 | . 2 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞))) |
23 | df-ss 3928 | . . . . 5 ⊢ ((ℤ≥‘𝑀) ⊆ ℤ ↔ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀)) | |
24 | 5, 23 | mpbi 229 | . . . 4 ⊢ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀) |
25 | 24 | ineq1i 4169 | . . 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 2785 | 1 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 ‘cfv 6497 (class class class)co 7358 +∞cpnf 11187 ℤcz 12500 ℤ≥cuz 12764 [,)cico 13267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-neg 11389 df-z 12501 df-uz 12765 df-ico 13271 |
This theorem is referenced by: uzinico3 43808 limsupvaluz 43956 |
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