xmetrtri - Intuitionistic Logic Explorer

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Theorem xmetrtri 15602

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 1017	. . 3 $/\$ $/\$ $/\$ $/\$
2		xmettri 15598	. . 3 $/\$ $/\$ $/\$
3	1, 2	sylan2b 287	. 2 $/\$ $/\$ $/\$
4		xmetcl 15578	. . . 4 $/\$ $/\$
5	4	3adant3r2 1244	. . 3 $/\$ $/\$ $/\$
6		xmetcl 15578	. . . 4 $/\$ $/\$
7	6	3adant3r1 1243	. . 3 $/\$ $/\$ $/\$
8		xmetcl 15578	. . . 4 $/\$ $/\$
9	8	3adant3r3 1245	. . 3 $/\$ $/\$ $/\$
10		xmetge0 15591	. . . 4 $/\$ $/\$
11	10	3adant3r2 1244	. . 3 $/\$ $/\$ $/\$
12		xmetge0 15591	. . . . 5 $/\$ $/\$
13	12	3adant3r1 1243	. . . 4 $/\$ $/\$ $/\$
14		ge0nemnf 10243	. . . 4 $/\$
15	7, 13, 14	syl2anc 415	. . 3 $/\$ $/\$ $/\$
16		xmetge0 15591	. . . 4 $/\$ $/\$
17	16	3adant3r3 1245	. . 3 $/\$ $/\$ $/\$
18		xlesubadd 10302	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
19	5, 7, 9, 11, 15, 17, 18	syl33anc 1293	. 2 $/\$ $/\$ $/\$
20	3, 19	mpbird 167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wcel 2209

wne 2420 class class class wbr 4131

cfv 5380 (class class class)co 6088

cc0 8185

cmnf 8364

cxr 8365

cle 8367

cxne 10188

cxad 10189

cxmet 14982

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-po 4443 df-iso 4444 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-map 6928 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-2 9372 df-xneg 10191 df-xadd 10192 df-xmet 14990

This theorem is used by: (None)

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