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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0infssd | Structured version Visualization version GIF version |
Description: Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
Ref | Expression |
---|---|
xrge0infssd.1 | ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
xrge0infssd.2 | ⊢ (𝜑 → 𝐵 ⊆ (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0infssd | ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12127 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 11850 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 4977 | . . . . . 6 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞))) | |
4 | 1, 2, 3 | mp2 9 | . . . . 5 ⊢ < Or (0[,]+∞) |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → < Or (0[,]+∞)) |
6 | xrge0infssd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (0[,]+∞)) | |
7 | xrge0infss 28915 | . . . . 5 ⊢ (𝐵 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐵 𝑧 < 𝑦))) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐵 𝑧 < 𝑦))) |
9 | 5, 8 | infcl 8277 | . . 3 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3566 | . 2 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ ℝ*) |
11 | xrge0infssd.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
12 | 11, 6 | sstrd 3578 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (0[,]+∞)) |
13 | xrge0infss 28915 | . . . . 5 ⊢ (𝐶 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐶 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐶 𝑧 < 𝑦))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐶 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐶 𝑧 < 𝑦))) |
15 | 5, 14 | infcl 8277 | . . 3 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ (0[,]+∞)) |
16 | 1, 15 | sseldi 3566 | . 2 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ ℝ*) |
17 | 5, 11, 14, 8 | infssd 28871 | . 2 ⊢ (𝜑 → ¬ inf(𝐶, (0[,]+∞), < ) < inf(𝐵, (0[,]+∞), < )) |
18 | 10, 16, 17 | xrnltled 9985 | 1 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 Or wor 4958 (class class class)co 6549 infcinf 8230 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 |
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-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
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-reu 2903 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-iun 4457 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-icc 12053 |
This theorem is referenced by: omsmon 29687 |
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