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 12916 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 12629 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5472 | . . . . . 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 30670 | . . . . 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 9037 | . . 3 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3885 | . 2 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ ℝ*) |
11 | xrge0infssd.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
12 | 11, 6 | sstrd 3897 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (0[,]+∞)) |
13 | xrge0infss 30670 | . . . . 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 9037 | . . 3 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ (0[,]+∞)) |
16 | 1, 15 | sseldi 3885 | . 2 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ ℝ*) |
17 | 5, 11, 14, 8 | infssd 30632 | . 2 ⊢ (𝜑 → ¬ inf(𝐶, (0[,]+∞), < ) < inf(𝐵, (0[,]+∞), < )) |
18 | 10, 16, 17 | xrnltled 10799 | 1 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wral 3054 ∃wrex 3055 ⊆ wss 3853 class class class wbr 5040 Or wor 5451 (class class class)co 7182 infcinf 8990 0cc0 10627 +∞cpnf 10762 ℝ*cxr 10764 < clt 10765 ≤ cle 10766 [,]cicc 12836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-icc 12840 |
This theorem is referenced by: omsmon 31847 |
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