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Theorem metdsle 24797

Description: The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)

**Hypothesis**
Ref	Expression
metdscn.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))

**Assertion**
Ref	Expression
metdsle	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐷,𝑦 𝑥,𝐵,𝑦 𝑥,𝑆,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

**Proof of Theorem metdsle**
Step	Hyp	Ref	Expression
1		simprr 772	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
2		simpr 484	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
3	2	sselda 3933	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑋)
4	3	adantrr 717	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
5	1, 4	jca 511	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
6		metdscn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
7	6	metdstri 24796	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)))
8	5, 7	syldan 591	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)))
9		simpll 766	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
10		xmetsym 24291	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵))
11	9, 1, 4, 10	syl3anc 1373	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵))
12	6	metds0 24795	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0)
13	12	3expa 1118	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0)
14	13	adantrr 717	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) = 0)
15	11, 14	oveq12d 7376	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +_𝑒 0))
16		xmetcl 24275	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ^*)
17	9, 4, 1, 16	syl3anc 1373	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ∈ ℝ^*)
18	17	xaddridd 13158	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) +_𝑒 0) = (𝐴𝐷𝐵))
19	15, 18	eqtrd 2771	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵))
20	8, 19	breqtrd 5124	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 ran crn 5625 ‘cfv 6492 (class class class)co 7358 infcinf 9344 0cc0 11026 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 +_𝑒 cxad 13024 ∞Metcxmet 21294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-ec 8637 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-2 12208 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-icc 13268 df-psmet 21301 df-xmet 21302 df-bl 21304

This theorem is referenced by: metdsre 24798

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