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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 13470 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 2 | xrltso 13183 | . . . . . 6 ⊢ < Or ℝ* | |
| 3 | soss 5612 | . . . . . 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 32764 | . . . . 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 9528 | . . 3 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ (0[,]+∞)) | 
| 10 | 1, 9 | sselid 3981 | . 2 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ∈ ℝ*) | 
| 11 | xrge0infssd.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
| 12 | 11, 6 | sstrd 3994 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (0[,]+∞)) | 
| 13 | xrge0infss 32764 | . . . . 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 9528 | . . 3 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ (0[,]+∞)) | 
| 16 | 1, 15 | sselid 3981 | . 2 ⊢ (𝜑 → inf(𝐶, (0[,]+∞), < ) ∈ ℝ*) | 
| 17 | 5, 11, 14, 8 | infssd 32722 | . 2 ⊢ (𝜑 → ¬ inf(𝐶, (0[,]+∞), < ) < inf(𝐵, (0[,]+∞), < )) | 
| 18 | 10, 16, 17 | xrnltled 11329 | 1 ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 Or wor 5591 (class class class)co 7431 infcinf 9481 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,]cicc 13390 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-icc 13394 | 
| This theorem is referenced by: omsmon 34300 | 
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