isxmet2d - Metamath Proof Explorer

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Theorem isxmet2d 22132

Description: It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample:

satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
isxmetd.0
isxmetd.1
isxmet2d.2	$/\$ $/\$
isxmet2d.3	$/\$ $/\$
isxmet2d.4	$/\$ $/\$ $/\$ $/\$ $/\$

**Assertion**
Ref	Expression
isxmet2d

Distinct variable groups:

**Proof of Theorem isxmet2d**
Step	Hyp	Ref	Expression
1		isxmetd.0	. 2
2		isxmetd.1	. 2
3	2	fovrnda 6805	. . . 4 $/\$ $/\$
4		0xr 10086	. . . 4
5		xrletri3 11985	. . . 4 $/\$ $/\$
6	3, 4, 5	sylancl 694	. . 3 $/\$ $/\$ $/\$
7		isxmet2d.2	. . . 4 $/\$ $/\$
8	7	biantrud 528	. . 3 $/\$ $/\$ $/\$
9		isxmet2d.3	. . 3 $/\$ $/\$
10	6, 8, 9	3bitr2d 296	. 2 $/\$ $/\$
11		isxmet2d.4	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	3expa 1265	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
13		rexadd 12063	. . . . . . 7 $/\$
14	13	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 4681	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 680	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1222	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 11964	. . . . . . 7
19	17, 18	syl 17	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 6658	. . . . . 6
22		ffn 6045	. . . . . . . . . . . 12
23	2, 22	syl 17	. . . . . . . . . . 11
24		elxrge0 12281	. . . . . . . . . . . . 13 $/\$
25	3, 7, 24	sylanbrc 698	. . . . . . . . . . . 12 $/\$ $/\$
26	25	ralrimivva 2971	. . . . . . . . . . 11
27		ffnov 6764	. . . . . . . . . . 11 $/\$
28	23, 26, 27	sylanbrc 698	. . . . . . . . . 10
29	28	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$
30		simpr3 1069	. . . . . . . . 9 $/\$ $/\$ $/\$
31		simpr1 1067	. . . . . . . . 9 $/\$ $/\$ $/\$
32	29, 30, 31	fovrnd 6806	. . . . . . . 8 $/\$ $/\$ $/\$
33		elxrge0 12281	. . . . . . . . 9 $/\$
34	33	simplbi 476	. . . . . . . 8
35	32, 34	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
36		renemnf 10088	. . . . . . 7
37		xaddpnf1 12057	. . . . . . 7 $/\$
38	35, 36, 37	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	21, 38	sylan9eqr 2678	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
40	20, 39	breqtrrd 4681	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
41		simpr2 1068	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42	29, 30, 41	fovrnd 6806	. . . . . . . . . 10 $/\$ $/\$ $/\$
43		elxrge0 12281	. . . . . . . . . . 11 $/\$
44	43	simplbi 476	. . . . . . . . . 10
45	42, 44	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$
46	43	simprbi 480	. . . . . . . . . 10
47	42, 46	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$
48		ge0nemnf 12004	. . . . . . . . 9 $/\$
49	45, 47, 48	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
50	49	a1d 25	. . . . . . 7 $/\$ $/\$ $/\$
51	50	necon4bd 2814	. . . . . 6 $/\$ $/\$ $/\$
52	51	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	52	imp 445	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
54	45	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$
55		elxr 11950	. . . . 5 $\/$ $\/$
56	54, 55	sylib 208	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
57	16, 40, 53, 56	mpjao3dan 1395	. . 3 $/\$ $/\$ $/\$ $/\$
58	19	adantr 481	. . . 4 $/\$ $/\$ $/\$ $/\$
59		oveq1 6657	. . . . 5
60		xaddpnf2 12058	. . . . . 6 $/\$
61	45, 49, 60	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$
62	59, 61	sylan9eqr 2678	. . . 4 $/\$ $/\$ $/\$ $/\$
63	58, 62	breqtrrd 4681	. . 3 $/\$ $/\$ $/\$ $/\$
64	33	simprbi 480	. . . . . . . 8
65	32, 64	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
66		ge0nemnf 12004	. . . . . . 7 $/\$
67	35, 65, 66	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
68	67	a1d 25	. . . . 5 $/\$ $/\$ $/\$
69	68	necon4bd 2814	. . . 4 $/\$ $/\$ $/\$
70	69	imp 445	. . 3 $/\$ $/\$ $/\$ $/\$
71		elxr 11950	. . . 4 $\/$ $\/$
72	35, 71	sylib 208	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
73	57, 63, 70, 72	mpjao3dan 1395	. 2 $/\$ $/\$ $/\$
74	1, 2, 10, 73	isxmetd 22131	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $\/$ w3o 1036 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

cvv 3200 class class class wbr 4653

cxp 5112

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cr 9935

cc0 9936

caddc 9939

cpnf 10071

cmnf 10072

cxr 10073

cle 10075

cxad 11944

cicc 12178

cxmt 19731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-xadd 11947 df-icc 12182 df-xmet 19739

This theorem is referenced by: prdsxmetlem 22173 xrsxmet 22612

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