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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 4228 | . . . 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 4209 | . . . . 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 12899 | . . . . . . . . 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 3984 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 9 | eqid 2737 | . . . . . . 7 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
| 10 | 8, 9 | uzinico 45573 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) = (ℤ ∩ (𝑁[,)+∞))) |
| 11 | 10 | eqcomd 2743 | . . . . 5 ⊢ (𝜑 → (ℤ ∩ (𝑁[,)+∞)) = (ℤ≥‘𝑁)) |
| 12 | 11 | ineq1d 4219 | . . . 4 ⊢ (𝜑 → ((ℤ ∩ (𝑁[,)+∞)) ∩ (ℤ≥‘𝑀)) = ((ℤ≥‘𝑁) ∩ (ℤ≥‘𝑀))) |
| 13 | 7 | uzssd 45419 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
| 14 | dfss2 3969 | . . . . 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 2781 | . . 3 ⊢ (𝜑 → ((ℤ≥‘𝑀) ∩ (ℤ ∩ (𝑁[,)+∞))) = (ℤ≥‘𝑁)) |
| 17 | uzssz 12899 | . . . . . 6 ⊢ (ℤ≥‘𝑁) ⊆ ℤ | |
| 18 | dfss2 3969 | . . . . . 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 2743 | . . 3 ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑁) ∩ ℤ)) |
| 22 | 2, 16, 21 | 3eqtrrd 2782 | . 2 ⊢ (𝜑 → ((ℤ≥‘𝑁) ∩ ℤ) = (((ℤ≥‘𝑀) ∩ ℤ) ∩ (𝑁[,)+∞))) |
| 23 | dfss2 3969 | . . . . 5 ⊢ ((ℤ≥‘𝑀) ⊆ ℤ ↔ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀)) | |
| 24 | 5, 23 | mpbi 230 | . . . 4 ⊢ ((ℤ≥‘𝑀) ∩ ℤ) = (ℤ≥‘𝑀) |
| 25 | 24 | ineq1i 4216 | . . 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 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 +∞cpnf 11292 ℤcz 12613 ℤ≥cuz 12878 [,)cico 13389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-ico 13393 |
| This theorem is referenced by: uzinico3 45576 limsupvaluz 45723 |
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