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Mirrors > Home > MPE Home > Th. List > metdscnlem | Structured version Visualization version GIF version |
Description: Lemma for metdscn 22467. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
metdscn.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
metdscn.c | ⊢ 𝐶 = (dist‘ℝ*𝑠) |
metdscn.k | ⊢ 𝐾 = (MetOpen‘𝐶) |
metdscnlem.1 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
metdscnlem.2 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
metdscnlem.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
metdscnlem.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
metdscnlem.5 | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
metdscnlem.6 | ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) |
Ref | Expression |
---|---|
metdscnlem | ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metdscnlem.1 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | metdscnlem.2 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
3 | metdscn.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
4 | 3 | metdsf 22459 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
5 | 1, 2, 4 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
6 | metdscnlem.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 5, 6 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (0[,]+∞)) |
8 | elxrge0 12152 | . . . . 5 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) ↔ ((𝐹‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐴))) | |
9 | 8 | simplbi 475 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → (𝐹‘𝐴) ∈ ℝ*) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ*) |
11 | metdscnlem.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
12 | 5, 11 | ffvelrnd 6268 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (0[,]+∞)) |
13 | elxrge0 12152 | . . . . . 6 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) ↔ ((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵))) | |
14 | 13 | simplbi 475 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ*) |
16 | 15 | xnegcld 12002 | . . 3 ⊢ (𝜑 → -𝑒(𝐹‘𝐵) ∈ ℝ*) |
17 | 10, 16 | xaddcld 12003 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ∈ ℝ*) |
18 | xmetcl 21946 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
19 | 1, 6, 11, 18 | syl3anc 1318 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) ∈ ℝ*) |
20 | metdscnlem.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
21 | 20 | rpxrd 11749 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
22 | 3 | metdstri 22462 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
23 | 1, 2, 6, 11, 22 | syl22anc 1319 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
24 | 8 | simprbi 479 | . . . . 5 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐴)) |
25 | 7, 24 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐴)) |
26 | 13 | simprbi 479 | . . . . . 6 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐵)) |
27 | 12, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐵)) |
28 | ge0nemnf 11878 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵)) → (𝐹‘𝐵) ≠ -∞) | |
29 | 15, 27, 28 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ≠ -∞) |
30 | xmetge0 21959 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | |
31 | 1, 6, 11, 30 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝐵)) |
32 | xlesubadd 11965 | . . . 4 ⊢ ((((𝐹‘𝐴) ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ* ∧ (𝐴𝐷𝐵) ∈ ℝ*) ∧ (0 ≤ (𝐹‘𝐴) ∧ (𝐹‘𝐵) ≠ -∞ ∧ 0 ≤ (𝐴𝐷𝐵))) → (((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))) | |
33 | 10, 15, 19, 25, 29, 31, 32 | syl33anc 1333 | . . 3 ⊢ (𝜑 → (((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))) |
34 | 23, 33 | mpbird 246 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
35 | metdscnlem.6 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) | |
36 | 17, 19, 21, 34, 35 | xrlelttrd 11867 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 infcinf 8230 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ℝ+crp 11708 -𝑒cxne 11819 +𝑒 cxad 11820 [,]cicc 12049 distcds 15777 ℝ*𝑠cxrs 15983 ∞Metcxmt 19552 MetOpencmopn 19557 |
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-ec 7631 df-map 7746 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-div 10564 df-2 10956 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-psmet 19559 df-xmet 19560 df-bl 19562 |
This theorem is referenced by: metdscn 22467 |
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