xmetge0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xmetge0		Structured version Visualization version GIF version

Theorem xmetge0 24263

Description: The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmetge0	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))

**Proof of Theorem xmetge0**
Step	Hyp	Ref	Expression
1		simp1 1134	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
2		simp2 1135	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1136	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
4		xmettri2 24259	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +_𝑒 (𝐴𝐷𝐵)))
5	1, 2, 3, 3, 4	syl13anc 1370	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +_𝑒 (𝐴𝐷𝐵)))
6		2re 12317	. . . . 5 ⊢ 2 ∈ ℝ
7		rexr 11291	. . . . 5 ⊢ (2 ∈ ℝ → 2 ∈ ℝ^*)
8		xmul01 13279	. . . . 5 ⊢ (2 ∈ ℝ^* → (2 ·_e 0) = 0)
9	6, 7, 8	mp2b 10	. . . 4 ⊢ (2 ·_e 0) = 0
10		xmet0 24261	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0)
11	10	3adant2 1129	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0)
12	9, 11	eqtr4id 2787	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·_e 0) = (𝐵𝐷𝐵))
13		xmetcl 24250	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ^*)
14		x2times 13311	. . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ^* → (2 ·_e (𝐴𝐷𝐵)) = ((𝐴𝐷𝐵) +_𝑒 (𝐴𝐷𝐵)))
15	13, 14	syl 17	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·_e (𝐴𝐷𝐵)) = ((𝐴𝐷𝐵) +_𝑒 (𝐴𝐷𝐵)))
16	5, 12, 15	3brtr4d 5180	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·_e 0) ≤ (2 ·_e (𝐴𝐷𝐵)))
17		0xr 11292	. . 3 ⊢ 0 ∈ ℝ^*
18		2rp 13012	. . . 4 ⊢ 2 ∈ ℝ⁺
19	18	a1i 11	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 2 ∈ ℝ⁺)
20		xlemul2 13303	. . 3 ⊢ ((0 ∈ ℝ^* ∧ (𝐴𝐷𝐵) ∈ ℝ^* ∧ 2 ∈ ℝ⁺) → (0 ≤ (𝐴𝐷𝐵) ↔ (2 ·_e 0) ≤ (2 ·_e (𝐴𝐷𝐵))))
21	17, 13, 19, 20	mp3an2i 1463	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 ≤ (𝐴𝐷𝐵) ↔ (2 ·_e 0) ≤ (2 ·_e (𝐴𝐷𝐵))))
22	16, 21	mpbird 257	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ℝcr 11138 0cc0 11139 ℝ^*cxr 11278 ≤ cle 11280 2c2 12298 ℝ⁺crp 13007 +_𝑒 cxad 13123 ·_e cxmu 13124 ∞Metcxmet 21264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-xmet 21272

This theorem is referenced by: metge0 24264 xmetlecl 24265 xmetrtri 24274 xmetgt0 24277 prdsxmetlem 24287 imasdsf1olem 24292 xpsdsval 24300 xblpnf 24315 blgt0 24318 xblss2 24321 xbln0 24333 xmsge0 24382 comet 24435 stdbdxmet 24437 stdbdmet 24438 xrsmopn 24741 metdsf 24777 metdstri 24780 metdscnlem 24784 iscfil2 25207 heicant 37128

W3C validator

	
		OSZAR »