xmetrtri - Intuitionistic Logic Explorer

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Theorem xmetrtri 13170

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 981	. . 3 $/\$ $/\$ $/\$ $/\$
2		xmettri 13166	. . 3 $/\$ $/\$ $/\$
3	1, 2	sylan2b 285	. 2 $/\$ $/\$ $/\$
4		xmetcl 13146	. . . 4 $/\$ $/\$
5	4	3adant3r2 1208	. . 3 $/\$ $/\$ $/\$
6		xmetcl 13146	. . . 4 $/\$ $/\$
7	6	3adant3r1 1207	. . 3 $/\$ $/\$ $/\$
8		xmetcl 13146	. . . 4 $/\$ $/\$
9	8	3adant3r3 1209	. . 3 $/\$ $/\$ $/\$
10		xmetge0 13159	. . . 4 $/\$ $/\$
11	10	3adant3r2 1208	. . 3 $/\$ $/\$ $/\$
12		xmetge0 13159	. . . . 5 $/\$ $/\$
13	12	3adant3r1 1207	. . . 4 $/\$ $/\$ $/\$
14		ge0nemnf 9781	. . . 4 $/\$
15	7, 13, 14	syl2anc 409	. . 3 $/\$ $/\$ $/\$
16		xmetge0 13159	. . . 4 $/\$ $/\$
17	16	3adant3r3 1209	. . 3 $/\$ $/\$ $/\$
18		xlesubadd 9840	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
19	5, 7, 9, 11, 15, 17, 18	syl33anc 1248	. 2 $/\$ $/\$ $/\$
20	3, 19	mpbird 166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 973

wcel 2141

wne 2340 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc0 7774

cmnf 7952

cxr 7953

cle 7955

cxne 9726

cxad 9727

cxmet 12774

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-2 8937 df-xneg 9729 df-xadd 9730 df-xmet 12782

This theorem is referenced by: (None)

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