isxmet2d - Intuitionistic Logic Explorer

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Theorem isxmet2d 15071

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	fovcdmda 6165	. . . 4 $/\$ $/\$
4		0xr 8225	. . . 4
5		xrletri3 10038	. . . 4 $/\$ $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$ $/\$ $/\$
7		isxmet2d.2	. . . 4 $/\$ $/\$
8	7	biantrud 304	. . 3 $/\$ $/\$ $/\$
9		isxmet2d.3	. . 3 $/\$ $/\$
10	6, 8, 9	3bitr2d 216	. 2 $/\$ $/\$
11		isxmet2d.4	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	3expa 1229	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
13		rexadd 10086	. . . . . . 7 $/\$
14	13	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 4116	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 400	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1184	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 10023	. . . . . . 7
19	17, 18	syl 14	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 6025	. . . . . 6
22	2	ffnd 5483	. . . . . . . . . . 11
23		elxrge0 10212	. . . . . . . . . . . . 13 $/\$
24	3, 7, 23	sylanbrc 417	. . . . . . . . . . . 12 $/\$ $/\$
25	24	ralrimivva 2614	. . . . . . . . . . 11
26		ffnov 6124	. . . . . . . . . . 11 $/\$
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10
28	27	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
29		simpr3 1031	. . . . . . . . 9 $/\$ $/\$ $/\$
30		simpr1 1029	. . . . . . . . 9 $/\$ $/\$ $/\$
31	28, 29, 30	fovcdmd 6166	. . . . . . . 8 $/\$ $/\$ $/\$
32		elxrge0 10212	. . . . . . . . 9 $/\$
33	32	simplbi 274	. . . . . . . 8
34	31, 33	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
35		renemnf 8227	. . . . . . 7
36		xaddpnf1 10080	. . . . . . 7 $/\$
37	34, 35, 36	syl2an 289	. . . . . 6 $/\$ $/\$ $/\$ $/\$
38	21, 37	sylan9eqr 2286	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
39	20, 38	breqtrrd 4116	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 1030	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	28, 29, 40	fovcdmd 6166	. . . . . . . . . 10 $/\$ $/\$ $/\$
42		elxrge0 10212	. . . . . . . . . . 11 $/\$
43	42	simplbi 274	. . . . . . . . . 10
44	41, 43	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
45	42	simprbi 275	. . . . . . . . . 10
46	41, 45	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
47		ge0nemnf 10058	. . . . . . . . 9 $/\$
48	44, 46, 47	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
49	48	neneqd 2423	. . . . . . 7 $/\$ $/\$ $/\$
50	49	pm2.21d 624	. . . . . 6 $/\$ $/\$ $/\$
51	50	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	51	imp 124	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	44	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
54		elxr 10010	. . . . 5 $\/$ $\/$
55	53, 54	sylib 122	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
56	16, 39, 52, 55	mpjao3dan 1343	. . 3 $/\$ $/\$ $/\$ $/\$
57	19	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
58		oveq1 6024	. . . . 5
59		xaddpnf2 10081	. . . . . 6 $/\$
60	44, 48, 59	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
61	58, 60	sylan9eqr 2286	. . . 4 $/\$ $/\$ $/\$ $/\$
62	57, 61	breqtrrd 4116	. . 3 $/\$ $/\$ $/\$ $/\$
63	32	simprbi 275	. . . . . . . 8
64	31, 63	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
65		ge0nemnf 10058	. . . . . . 7 $/\$
66	34, 64, 65	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
67	66	neneqd 2423	. . . . 5 $/\$ $/\$ $/\$
68	67	pm2.21d 624	. . . 4 $/\$ $/\$ $/\$
69	68	imp 124	. . 3 $/\$ $/\$ $/\$ $/\$
70		elxr 10010	. . . 4 $\/$ $\/$
71	34, 70	sylib 122	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
72	56, 62, 69, 71	mpjao3dan 1343	. 2 $/\$ $/\$ $/\$
73	1, 2, 10, 72	isxmetd 15070	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ w3o 1003 $/\$ w3a 1004

wceq 1397

wcel 2202

wne 2402

wral 2510

cvv 2802 class class class wbr 4088

cxp 4723

wfn 5321

wf 5322

cfv 5326 (class class class)co 6017

cr 8030

cc0 8031

caddc 8034

cpnf 8210

cmnf 8211

cxr 8212

cle 8214

cxad 10004

cicc 10125

cxmet 14549

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1re 8125 ax-addrcl 8128 ax-rnegex 8140 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-map 6818 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-xadd 10007 df-icc 10129 df-xmet 14557

This theorem is referenced by: (None)

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