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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfgval | Structured version Visualization version GIF version |
Description: Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfgval.1 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Ref | Expression |
---|---|
liminfgval | ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7166 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘[,)+∞) = (𝑀[,)+∞)) | |
2 | 1 | imaeq2d 5932 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞))) |
3 | 2 | ineq1d 4191 | . . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*)) |
4 | 3 | infeq1d 8944 | . 2 ⊢ (𝑘 = 𝑀 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
5 | liminfgval.1 | . 2 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
6 | xrltso 12537 | . . 3 ⊢ < Or ℝ* | |
7 | 6 | infex 8960 | . 2 ⊢ inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V |
8 | 4, 5, 7 | fvmpt 6771 | 1 ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ↦ cmpt 5149 “ cima 5561 ‘cfv 6358 (class class class)co 7159 infcinf 8908 ℝcr 10539 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 [,)cico 12743 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 |
This theorem is referenced by: liminfval2 42055 |
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