![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > metdscnlem | Structured version Visualization version GIF version |
Description: Lemma for metdscn 24892. (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 24884 | . . . . . 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 7105 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (0[,]+∞)) |
8 | eliccxr 13472 | . . . 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 7105 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (0[,]+∞)) |
12 | eliccxr 13472 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ*) |
14 | 13 | xnegcld 13339 | . . 3 ⊢ (𝜑 → -𝑒(𝐹‘𝐵) ∈ ℝ*) |
15 | 9, 14 | xaddcld 13340 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ∈ ℝ*) |
16 | xmetcl 24357 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
17 | 1, 6, 10, 16 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) ∈ ℝ*) |
18 | metdscnlem.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
19 | 18 | rpxrd 13076 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
20 | 3 | metdstri 24887 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
21 | 1, 2, 6, 10, 20 | syl22anc 839 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) |
22 | elxrge0 13494 | . . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) ↔ ((𝐹‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐴))) | |
23 | 22 | simprbi 496 | . . . . 5 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐴)) |
24 | 7, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐴)) |
25 | elxrge0 13494 | . . . . . . 7 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) ↔ ((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵))) | |
26 | 25 | simprbi 496 | . . . . . 6 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐵)) |
27 | 11, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐵)) |
28 | ge0nemnf 13212 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝐵)) → (𝐹‘𝐵) ≠ -∞) | |
29 | 13, 27, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ≠ -∞) |
30 | xmetge0 24370 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | |
31 | 1, 6, 10, 30 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝐵)) |
32 | xlesubadd 13302 | . . . 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 1384 | . . 3 ⊢ (𝜑 → (((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))) |
34 | 21, 33 | mpbird 257 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
35 | metdscnlem.6 | . 2 ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) | |
36 | 15, 17, 19, 34, 35 | xrlelttrd 13199 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 infcinf 9479 0cc0 11153 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 ℝ+crp 13032 -𝑒cxne 13149 +𝑒 cxad 13150 [,]cicc 13387 distcds 17307 ℝ*𝑠cxrs 17547 ∞Metcxmet 21367 MetOpencmopn 21372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-ec 8746 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-psmet 21374 df-xmet 21375 df-bl 21377 |
This theorem is referenced by: metdscn 24892 |
Copyright terms: Public domain | W3C validator |