xmetrtri - Intuitionistic Logic Explorer

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Theorem xmetrtri 15050

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 1010	. . 3 $/\$ $/\$ $/\$ $/\$
2		xmettri 15046	. . 3 $/\$ $/\$ $/\$
3	1, 2	sylan2b 287	. 2 $/\$ $/\$ $/\$
4		xmetcl 15026	. . . 4 $/\$ $/\$
5	4	3adant3r2 1237	. . 3 $/\$ $/\$ $/\$
6		xmetcl 15026	. . . 4 $/\$ $/\$
7	6	3adant3r1 1236	. . 3 $/\$ $/\$ $/\$
8		xmetcl 15026	. . . 4 $/\$ $/\$
9	8	3adant3r3 1238	. . 3 $/\$ $/\$ $/\$
10		xmetge0 15039	. . . 4 $/\$ $/\$
11	10	3adant3r2 1237	. . 3 $/\$ $/\$ $/\$
12		xmetge0 15039	. . . . 5 $/\$ $/\$
13	12	3adant3r1 1236	. . . 4 $/\$ $/\$ $/\$
14		ge0nemnf 10020	. . . 4 $/\$
15	7, 13, 14	syl2anc 411	. . 3 $/\$ $/\$ $/\$
16		xmetge0 15039	. . . 4 $/\$ $/\$
17	16	3adant3r3 1238	. . 3 $/\$ $/\$ $/\$
18		xlesubadd 10079	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
19	5, 7, 9, 11, 15, 17, 18	syl33anc 1286	. 2 $/\$ $/\$ $/\$
20	3, 19	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wcel 2200

wne 2400 class class class wbr 4083

cfv 5318 (class class class)co 6001

cc0 7999

cmnf 8179

cxr 8180

cle 8182

cxne 9965

cxad 9966

cxmet 14500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-map 6797 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-2 9169 df-xneg 9968 df-xadd 9969 df-xmet 14508

This theorem is referenced by: (None)

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