isxmet2d - Intuitionistic Logic Explorer

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Theorem isxmet2d 15159

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 6176	. . . 4 $/\$ $/\$
4		0xr 8285	. . . 4
5		xrletri3 10100	. . . 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 10148	. . . . . . 7 $/\$
14	13	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 4121	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 400	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1185	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 10085	. . . . . . 7
19	17, 18	syl 14	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 6036	. . . . . 6
22	2	ffnd 5490	. . . . . . . . . . 11
23		elxrge0 10274	. . . . . . . . . . . . 13 $/\$
24	3, 7, 23	sylanbrc 417	. . . . . . . . . . . 12 $/\$ $/\$
25	24	ralrimivva 2615	. . . . . . . . . . 11
26		ffnov 6135	. . . . . . . . . . 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 6177	. . . . . . . 8 $/\$ $/\$ $/\$
32		elxrge0 10274	. . . . . . . . 9 $/\$
33	32	simplbi 274	. . . . . . . 8
34	31, 33	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
35		renemnf 8287	. . . . . . 7
36		xaddpnf1 10142	. . . . . . 7 $/\$
37	34, 35, 36	syl2an 289	. . . . . 6 $/\$ $/\$ $/\$ $/\$
38	21, 37	sylan9eqr 2286	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
39	20, 38	breqtrrd 4121	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	28, 29, 40	fovcdmd 6177	. . . . . . . . . 10 $/\$ $/\$ $/\$
42		elxrge0 10274	. . . . . . . . . . 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 10120	. . . . . . . . 9 $/\$
48	44, 46, 47	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
49	48	neneqd 2424	. . . . . . 7 $/\$ $/\$ $/\$
50	49	pm2.21d 624	. . . . . 6 $/\$ $/\$ $/\$
51	50	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	51	imp 124	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	44	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
54		elxr 10072	. . . . 5 $\/$ $\/$
55	53, 54	sylib 122	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
56	16, 39, 52, 55	mpjao3dan 1344	. . 3 $/\$ $/\$ $/\$ $/\$
57	19	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
58		oveq1 6035	. . . . 5
59		xaddpnf2 10143	. . . . . 6 $/\$
60	44, 48, 59	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
61	58, 60	sylan9eqr 2286	. . . 4 $/\$ $/\$ $/\$ $/\$
62	57, 61	breqtrrd 4121	. . 3 $/\$ $/\$ $/\$ $/\$
63	32	simprbi 275	. . . . . . . 8
64	31, 63	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
65		ge0nemnf 10120	. . . . . . 7 $/\$
66	34, 64, 65	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
67	66	neneqd 2424	. . . . 5 $/\$ $/\$ $/\$
68	67	pm2.21d 624	. . . 4 $/\$ $/\$ $/\$
69	68	imp 124	. . 3 $/\$ $/\$ $/\$ $/\$
70		elxr 10072	. . . 4 $\/$ $\/$
71	34, 70	sylib 122	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
72	56, 62, 69, 71	mpjao3dan 1344	. 2 $/\$ $/\$ $/\$
73	1, 2, 10, 72	isxmetd 15158	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

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

wceq 1398

wcel 2202

wne 2403

wral 2511

cvv 2803 class class class wbr 4093

cxp 4729

wfn 5328

wf 5329

cfv 5333 (class class class)co 6028

cr 8091

cc0 8092

caddc 8095

cpnf 8270

cmnf 8271

cxr 8272

cle 8274

cxad 10066

cicc 10187

cxmet 14632

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1re 8186 ax-addrcl 8189 ax-rnegex 8201 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-po 4399 df-iso 4400 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-map 6862 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-xadd 10069 df-icc 10191 df-xmet 14640

This theorem is referenced by: (None)

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