isxmet2d - Intuitionistic Logic Explorer

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Theorem isxmet2d 15339

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 6206	. . . 4 $/\$ $/\$
4		0xr 8336	. . . 4
5		xrletri3 10156	. . . 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 1230	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
13		rexadd 10204	. . . . . . 7 $/\$
14	13	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 4142	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 400	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1185	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 10141	. . . . . . 7
19	17, 18	syl 14	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 6066	. . . . . 6
22	2	ffnd 5514	. . . . . . . . . . 11
23		elxrge0 10330	. . . . . . . . . . . . 13 $/\$
24	3, 7, 23	sylanbrc 417	. . . . . . . . . . . 12 $/\$ $/\$
25	24	ralrimivva 2626	. . . . . . . . . . 11
26		ffnov 6165	. . . . . . . . . . 11 $/\$
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10
28	27	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
29		simpr3 1032	. . . . . . . . 9 $/\$ $/\$ $/\$
30		simpr1 1030	. . . . . . . . 9 $/\$ $/\$ $/\$
31	28, 29, 30	fovcdmd 6207	. . . . . . . 8 $/\$ $/\$ $/\$
32		elxrge0 10330	. . . . . . . . 9 $/\$
33	32	simplbi 274	. . . . . . . 8
34	31, 33	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
35		renemnf 8338	. . . . . . 7
36		xaddpnf1 10198	. . . . . . 7 $/\$
37	34, 35, 36	syl2an 289	. . . . . 6 $/\$ $/\$ $/\$ $/\$
38	21, 37	sylan9eqr 2289	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
39	20, 38	breqtrrd 4142	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	28, 29, 40	fovcdmd 6207	. . . . . . . . . 10 $/\$ $/\$ $/\$
42		elxrge0 10330	. . . . . . . . . . 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 10176	. . . . . . . . 9 $/\$
48	44, 46, 47	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
49	48	neneqd 2435	. . . . . . 7 $/\$ $/\$ $/\$
50	49	pm2.21d 624	. . . . . 6 $/\$ $/\$ $/\$
51	50	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	51	imp 124	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	44	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
54		elxr 10128	. . . . 5 $\/$ $\/$
55	53, 54	sylib 122	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
56	16, 39, 52, 55	mpjao3dan 1344	. . 3 $/\$ $/\$ $/\$ $/\$
57	19	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
58		oveq1 6065	. . . . 5
59		xaddpnf2 10199	. . . . . 6 $/\$
60	44, 48, 59	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
61	58, 60	sylan9eqr 2289	. . . 4 $/\$ $/\$ $/\$ $/\$
62	57, 61	breqtrrd 4142	. . 3 $/\$ $/\$ $/\$ $/\$
63	32	simprbi 275	. . . . . . . 8
64	31, 63	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
65		ge0nemnf 10176	. . . . . . 7 $/\$
66	34, 64, 65	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
67	66	neneqd 2435	. . . . 5 $/\$ $/\$ $/\$
68	67	pm2.21d 624	. . . 4 $/\$ $/\$ $/\$
69	68	imp 124	. . 3 $/\$ $/\$ $/\$ $/\$
70		elxr 10128	. . . 4 $\/$ $\/$
71	34, 70	sylib 122	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
72	56, 62, 69, 71	mpjao3dan 1344	. 2 $/\$ $/\$ $/\$
73	1, 2, 10, 72	isxmetd 15338	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ w3o 1004 $/\$ w3a 1005

wceq 1398

wcel 2205

wne 2414

wral 2522

cvv 2815 class class class wbr 4114

cxp 4752

wfn 5352

wf 5353

cfv 5357 (class class class)co 6058

cr 8142

cc0 8143

caddc 8146

cpnf 8321

cmnf 8322

cxr 8323

cle 8325

cxad 10122

cicc 10243

cxmet 14810

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1re 8237 ax-addrcl 8240 ax-rnegex 8252 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-map 6897 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-xadd 10125 df-icc 10247 df-xmet 14818

This theorem is referenced by: (None)

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