metdscnlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > metdscnlem		Structured version Visualization version GIF version

Theorem metdscnlem 23457

Description: Lemma for metdscn 23458. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
metdscnlem	⊢ (𝜑 → ((𝐹‘𝐴) +_𝑒 -_𝑒(𝐹‘𝐵)) < 𝑅)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐷,𝑦 𝑦,𝐽 𝑥,𝐵,𝑦 𝑥,𝑆,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐶(𝑥,𝑦) 𝑅(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐽(𝑥) 𝐾(𝑥,𝑦)

**Proof of Theorem metdscnlem**
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 23450	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
5	1, 2, 4	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
6		metdscnlem.3	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7	5, 6	ffvelrnd 6847	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ (0[,]+∞))
8		eliccxr 12817	. . . 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	ffvelrnd 6847	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (0[,]+∞))
12		eliccxr 12817	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ^*)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ^*)
14	13	xnegcld 12687	. . 3 ⊢ (𝜑 → -_𝑒(𝐹‘𝐵) ∈ ℝ^*)
15	9, 14	xaddcld 12688	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) +_𝑒 -_𝑒(𝐹‘𝐵)) ∈ ℝ^*)
16		xmetcl 22935	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ^*)
17	1, 6, 10, 16	syl3anc 1367	. 2 ⊢ (𝜑 → (𝐴𝐷𝐵) ∈ ℝ^*)
18		metdscnlem.5	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
19	18	rpxrd 12426	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
20	3	metdstri 23453	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +_𝑒 (𝐹‘𝐵)))
21	1, 2, 6, 10, 20	syl22anc 836	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +_𝑒 (𝐹‘𝐵)))
22		elxrge0 12839	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) ↔ ((𝐹‘𝐴) ∈ ℝ^* ∧ 0 ≤ (𝐹‘𝐴)))
23	22	simprbi 499	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐴))
24	7, 23	syl 17	. . . 4 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐴))
25		elxrge0 12839	. . . . . . 7 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) ↔ ((𝐹‘𝐵) ∈ ℝ^* ∧ 0 ≤ (𝐹‘𝐵)))
26	25	simprbi 499	. . . . . 6 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐵))
27	11, 26	syl 17	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝐵))
28		ge0nemnf 12560	. . . . 5 ⊢ (((𝐹‘𝐵) ∈ ℝ^* ∧ 0 ≤ (𝐹‘𝐵)) → (𝐹‘𝐵) ≠ -∞)
29	13, 27, 28	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ≠ -∞)
30		xmetge0 22948	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
31	1, 6, 10, 30	syl3anc 1367	. . . 4 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝐵))
32		xlesubadd 12650	. . . 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 1381	. . 3 ⊢ (𝜑 → (((𝐹‘𝐴) +_𝑒 -_𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵) ↔ (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +_𝑒 (𝐹‘𝐵))))
34	21, 33	mpbird 259	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) +_𝑒 -_𝑒(𝐹‘𝐵)) ≤ (𝐴𝐷𝐵))
35		metdscnlem.6	. 2 ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅)
36	15, 17, 19, 34, 35	xrlelttrd 12547	1 ⊢ (𝜑 → ((𝐹‘𝐴) +_𝑒 -_𝑒(𝐹‘𝐵)) < 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3936 class class class wbr 5059 ↦ cmpt 5139 ran crn 5551 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 infcinf 8899 0cc0 10531 +∞cpnf 10666 -∞cmnf 10667 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 ℝ⁺crp 12383 -_𝑒cxne 12498 +_𝑒 cxad 12499 [,]cicc 12735 distcds 16568 ℝ^*_𝑠cxrs 16767 ∞Metcxmet 20524 MetOpencmopn 20529

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-ec 8285 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-psmet 20531 df-xmet 20532 df-bl 20534

This theorem is referenced by: metdscn 23458

W3C validator