![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 13353 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 13066 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5566 | . . . . . 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 31712 | . . . . 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 9429 | . . 3 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3943 | . 2 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ ℝ*) |
11 | xrge0infssd.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
12 | 11, 6 | sstrd 3955 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (0[,]+∞)) |
13 | xrge0infss 31712 | . . . . 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 9429 | . . 3 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ (0[,]+∞)) |
16 | 1, 15 | sselid 3943 | . 2 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ ℝ*) |
17 | 5, 11, 14, 8 | infssd 31674 | . 2 ⊢ (𝜑 → ¬ inf(𝐶, (0[,]+∞), < ) < inf(𝐵, (0[,]+∞), < )) |
18 | 10, 16, 17 | xrnltled 11228 | 1 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∀wral 3061 ∃wrex 3070 ⊆ wss 3911 class class class wbr 5106 Or wor 5545 (class class class)co 7358 infcinf 9382 0cc0 11056 +∞cpnf 11191 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 [,]cicc 13273 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 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-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-icc 13277 |
This theorem is referenced by: omsmon 32955 |
Copyright terms: Public domain | W3C validator |