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Mathbox for Glauco Siliprandi |
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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 12780 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | 2 | uzinico2 43807 | . 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 4172 | . . 3 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑀[,)+∞))) |
7 | 5, 6 | eqeq12d 2753 | . 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 1542 ∈ wcel 2107 ∩ cin 3910 ‘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: liminfvaluz 44040 limsupvaluz3 44046 |
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