xmetrtri - Intuitionistic Logic Explorer

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Theorem xmetrtri 12917

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 975	. . 3 $/\$ $/\$ $/\$ $/\$
2		xmettri 12913	. . 3 $/\$ $/\$ $/\$
3	1, 2	sylan2b 285	. 2 $/\$ $/\$ $/\$
4		xmetcl 12893	. . . 4 $/\$ $/\$
5	4	3adant3r2 1202	. . 3 $/\$ $/\$ $/\$
6		xmetcl 12893	. . . 4 $/\$ $/\$
7	6	3adant3r1 1201	. . 3 $/\$ $/\$ $/\$
8		xmetcl 12893	. . . 4 $/\$ $/\$
9	8	3adant3r3 1203	. . 3 $/\$ $/\$ $/\$
10		xmetge0 12906	. . . 4 $/\$ $/\$
11	10	3adant3r2 1202	. . 3 $/\$ $/\$ $/\$
12		xmetge0 12906	. . . . 5 $/\$ $/\$
13	12	3adant3r1 1201	. . . 4 $/\$ $/\$ $/\$
14		ge0nemnf 9751	. . . 4 $/\$
15	7, 13, 14	syl2anc 409	. . 3 $/\$ $/\$ $/\$
16		xmetge0 12906	. . . 4 $/\$ $/\$
17	16	3adant3r3 1203	. . 3 $/\$ $/\$ $/\$
18		xlesubadd 9810	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
19	5, 7, 9, 11, 15, 17, 18	syl33anc 1242	. 2 $/\$ $/\$ $/\$
20	3, 19	mpbird 166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 967

wcel 2135

wne 2334 class class class wbr 3976

cfv 5182 (class class class)co 5836

cc0 7744

cmnf 7922

cxr 7923

cle 7925

cxne 9696

cxad 9697

cxmet 12521

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-2 8907 df-xneg 9699 df-xadd 9700 df-xmet 12529

This theorem is referenced by: (None)

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