![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > metdscnlem | Structured version Visualization version GIF version |
Description: Lemma for metdscn 22706. (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 22698 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
5 | 1, 2, 4 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
6 | metdscnlem.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 5, 6 | ffvelrnd 6400 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (0[,]+∞)) |
8 | elxrge0 12319 | . . . . 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 6400 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (0[,]+∞)) |
13 | elxrge0 12319 | . . . . . 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 12168 | . . 3 ⊢ (𝜑 → -𝑒(𝐹‘𝐵) ∈ ℝ*) |
17 | 10, 16 | xaddcld 12169 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ∈ ℝ*) |
18 | xmetcl 22183 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
19 | 1, 6, 11, 18 | syl3anc 1366 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) ∈ ℝ*) |
20 | metdscnlem.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
21 | 20 | rpxrd 11911 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
22 | 3 | metdstri 22701 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
23 | 1, 2, 6, 11, 22 | syl22anc 1367 | . . 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 12042 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵)) → (𝐹‘𝐵) ≠ -∞) | |
29 | 15, 27, 28 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ≠ -∞) |
30 | xmetge0 22196 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | |
31 | 1, 6, 11, 30 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝐵)) |
32 | xlesubadd 12131 | . . . 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 1381 | . . 3 ⊢ (𝜑 → (((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))) |
34 | 23, 33 | mpbird 247 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
35 | metdscnlem.6 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) | |
36 | 17, 19, 21, 34, 35 | xrlelttrd 12029 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ⊆ wss 3607 class class class wbr 4685 ↦ cmpt 4762 ran crn 5144 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 infcinf 8388 0cc0 9974 +∞cpnf 10109 -∞cmnf 10110 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 ℝ+crp 11870 -𝑒cxne 11981 +𝑒 cxad 11982 [,]cicc 12216 distcds 15997 ℝ*𝑠cxrs 16207 ∞Metcxmt 19779 MetOpencmopn 19784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-ec 7789 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-2 11117 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-icc 12220 df-psmet 19786 df-xmet 19787 df-bl 19789 |
This theorem is referenced by: metdscn 22706 |
Copyright terms: Public domain | W3C validator |