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Mirrors > Home > MPE Home > Th. List > limsupgval | Structured version Visualization version GIF version |
Description: Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) |
Ref | Expression |
---|---|
limsupval.1 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Ref | Expression |
---|---|
limsupgval | ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6556 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘[,)+∞) = (𝑀[,)+∞)) | |
2 | 1 | imaeq2d 5385 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞))) |
3 | 2 | ineq1d 3775 | . . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*)) |
4 | 3 | supeq1d 8235 | . 2 ⊢ (𝑘 = 𝑀 → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
5 | limsupval.1 | . 2 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
6 | xrltso 11850 | . . 3 ⊢ < Or ℝ* | |
7 | 6 | supex 8252 | . 2 ⊢ sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V |
8 | 4, 5, 7 | fvmpt 6191 | 1 ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 |
This theorem is referenced by: limsupgle 14056 limsupval2 14059 limsupgre 14060 |
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