xmetrtri - Intuitionistic Logic Explorer

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Theorem xmetrtri 12550

Description: One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Assertion**
Ref	Expression
xmetrtri	$/\$ $/\$ $/\$

**Proof of Theorem xmetrtri**
Step	Hyp	Ref	Expression
1		3ancomb 970	. . 3 $/\$ $/\$ $/\$ $/\$
2		xmettri 12546	. . 3 $/\$ $/\$ $/\$
3	1, 2	sylan2b 285	. 2 $/\$ $/\$ $/\$
4		xmetcl 12526	. . . 4 $/\$ $/\$
5	4	3adant3r2 1191	. . 3 $/\$ $/\$ $/\$
6		xmetcl 12526	. . . 4 $/\$ $/\$
7	6	3adant3r1 1190	. . 3 $/\$ $/\$ $/\$
8		xmetcl 12526	. . . 4 $/\$ $/\$
9	8	3adant3r3 1192	. . 3 $/\$ $/\$ $/\$
10		xmetge0 12539	. . . 4 $/\$ $/\$
11	10	3adant3r2 1191	. . 3 $/\$ $/\$ $/\$
12		xmetge0 12539	. . . . 5 $/\$ $/\$
13	12	3adant3r1 1190	. . . 4 $/\$ $/\$ $/\$
14		ge0nemnf 9612	. . . 4 $/\$
15	7, 13, 14	syl2anc 408	. . 3 $/\$ $/\$ $/\$
16		xmetge0 12539	. . . 4 $/\$ $/\$
17	16	3adant3r3 1192	. . 3 $/\$ $/\$ $/\$
18		xlesubadd 9671	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
19	5, 7, 9, 11, 15, 17, 18	syl33anc 1231	. 2 $/\$ $/\$ $/\$
20	3, 19	mpbird 166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 962

wcel 1480

wne 2308 class class class wbr 3929

cfv 5123 (class class class)co 5774

cc0 7625

cmnf 7803

cxr 7804

cle 7806

cxne 9561

cxad 9562

cxmet 12154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-mulrcl 7724 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-precex 7735 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 ax-pre-mulgt0 7742

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-2 8784 df-xneg 9564 df-xadd 9565 df-xmet 12162

This theorem is referenced by: (None)

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