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Mirrors > Home > MPE Home > Th. List > metdscnlem | Structured version Visualization version GIF version |
Description: Lemma for metdscn 24219. (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 24211 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
5 | 1, 2, 4 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
6 | metdscnlem.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 5, 6 | ffvelcdmd 7036 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (0[,]+∞)) |
8 | eliccxr 13352 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → (𝐹‘𝐴) ∈ ℝ*) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ*) |
10 | metdscnlem.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
11 | 5, 10 | ffvelcdmd 7036 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (0[,]+∞)) |
12 | eliccxr 13352 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ*) |
14 | 13 | xnegcld 13219 | . . 3 ⊢ (𝜑 → -𝑒(𝐹‘𝐵) ∈ ℝ*) |
15 | 9, 14 | xaddcld 13220 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ∈ ℝ*) |
16 | xmetcl 23684 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
17 | 1, 6, 10, 16 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) ∈ ℝ*) |
18 | metdscnlem.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
19 | 18 | rpxrd 12958 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
20 | 3 | metdstri 24214 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
21 | 1, 2, 6, 10, 20 | syl22anc 837 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
22 | elxrge0 13374 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) ↔ ((𝐹‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐴))) | |
23 | 22 | simprbi 497 | . . . . 5 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐴)) |
24 | 7, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐴)) |
25 | elxrge0 13374 | . . . . . . 7 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) ↔ ((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵))) | |
26 | 25 | simprbi 497 | . . . . . 6 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐵)) |
27 | 11, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐵)) |
28 | ge0nemnf 13092 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵)) → (𝐹‘𝐵) ≠ -∞) | |
29 | 13, 27, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ≠ -∞) |
30 | xmetge0 23697 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | |
31 | 1, 6, 10, 30 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝐵)) |
32 | xlesubadd 13182 | . . . 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 | 9, 13, 17, 24, 29, 31, 32 | syl33anc 1385 | . . 3 ⊢ (𝜑 → (((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))) |
34 | 21, 33 | mpbird 256 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
35 | metdscnlem.6 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) | |
36 | 15, 17, 19, 34, 35 | xrlelttrd 13079 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 infcinf 9377 0cc0 11051 +∞cpnf 11186 -∞cmnf 11187 ℝ*cxr 11188 < clt 11189 ≤ cle 11190 ℝ+crp 12915 -𝑒cxne 13030 +𝑒 cxad 13031 [,]cicc 13267 distcds 17142 ℝ*𝑠cxrs 17382 ∞Metcxmet 20781 MetOpencmopn 20786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-er 8648 df-ec 8650 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-2 12216 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-psmet 20788 df-xmet 20789 df-bl 20791 |
This theorem is referenced by: metdscn 24219 |
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