isxmet2d - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > isxmet2d		Unicode version

Theorem isxmet2d 12517

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 5914	. . . 4 $/\$ $/\$
4		0xr 7812	. . . 4
5		xrletri3 9588	. . . 4 $/\$ $/\$
6	3, 4, 5	sylancl 409	. . 3 $/\$ $/\$ $/\$
7		isxmet2d.2	. . . 4 $/\$ $/\$
8	7	biantrud 302	. . 3 $/\$ $/\$ $/\$
9		isxmet2d.3	. . 3 $/\$ $/\$
10	6, 8, 9	3bitr2d 215	. 2 $/\$ $/\$
11		isxmet2d.4	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	3expa 1181	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
13		rexadd 9635	. . . . . . 7 $/\$
14	13	adantl 275	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 3956	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 397	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1142	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 9575	. . . . . . 7
19	17, 18	syl 14	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 479	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 5782	. . . . . 6
22	2	ffnd 5273	. . . . . . . . . . 11
23		elxrge0 9761	. . . . . . . . . . . . 13 $/\$
24	3, 7, 23	sylanbrc 413	. . . . . . . . . . . 12 $/\$ $/\$
25	24	ralrimivva 2514	. . . . . . . . . . 11
26		ffnov 5875	. . . . . . . . . . 11 $/\$
27	22, 25, 26	sylanbrc 413	. . . . . . . . . 10
28	27	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
29		simpr3 989	. . . . . . . . 9 $/\$ $/\$ $/\$
30		simpr1 987	. . . . . . . . 9 $/\$ $/\$ $/\$
31	28, 29, 30	fovrnd 5915	. . . . . . . 8 $/\$ $/\$ $/\$
32		elxrge0 9761	. . . . . . . . 9 $/\$
33	32	simplbi 272	. . . . . . . 8
34	31, 33	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
35		renemnf 7814	. . . . . . 7
36		xaddpnf1 9629	. . . . . . 7 $/\$
37	34, 35, 36	syl2an 287	. . . . . 6 $/\$ $/\$ $/\$ $/\$
38	21, 37	sylan9eqr 2194	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
39	20, 38	breqtrrd 3956	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 988	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	28, 29, 40	fovrnd 5915	. . . . . . . . . 10 $/\$ $/\$ $/\$
42		elxrge0 9761	. . . . . . . . . . 11 $/\$
43	42	simplbi 272	. . . . . . . . . 10
44	41, 43	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
45	42	simprbi 273	. . . . . . . . . 10
46	41, 45	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
47		ge0nemnf 9607	. . . . . . . . 9 $/\$
48	44, 46, 47	syl2anc 408	. . . . . . . 8 $/\$ $/\$ $/\$
49	48	neneqd 2329	. . . . . . 7 $/\$ $/\$ $/\$
50	49	pm2.21d 608	. . . . . 6 $/\$ $/\$ $/\$
51	50	adantr 274	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	51	imp 123	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	44	adantr 274	. . . . 5 $/\$ $/\$ $/\$ $/\$
54		elxr 9563	. . . . 5 $\/$ $\/$
55	53, 54	sylib 121	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
56	16, 39, 52, 55	mpjao3dan 1285	. . 3 $/\$ $/\$ $/\$ $/\$
57	19	adantr 274	. . . 4 $/\$ $/\$ $/\$ $/\$
58		oveq1 5781	. . . . 5
59		xaddpnf2 9630	. . . . . 6 $/\$
60	44, 48, 59	syl2anc 408	. . . . 5 $/\$ $/\$ $/\$
61	58, 60	sylan9eqr 2194	. . . 4 $/\$ $/\$ $/\$ $/\$
62	57, 61	breqtrrd 3956	. . 3 $/\$ $/\$ $/\$ $/\$
63	32	simprbi 273	. . . . . . . 8
64	31, 63	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
65		ge0nemnf 9607	. . . . . . 7 $/\$
66	34, 64, 65	syl2anc 408	. . . . . 6 $/\$ $/\$ $/\$
67	66	neneqd 2329	. . . . 5 $/\$ $/\$ $/\$
68	67	pm2.21d 608	. . . 4 $/\$ $/\$ $/\$
69	68	imp 123	. . 3 $/\$ $/\$ $/\$ $/\$
70		elxr 9563	. . . 4 $\/$ $\/$
71	34, 70	sylib 121	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
72	56, 62, 69, 71	mpjao3dan 1285	. 2 $/\$ $/\$ $/\$
73	1, 2, 10, 72	isxmetd 12516	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ w3o 961 $/\$ w3a 962

wceq 1331

wcel 1480

wne 2308

wral 2416

cvv 2686 class class class wbr 3929

cxp 4537

wfn 5118

wf 5119

cfv 5123 (class class class)co 5774

cr 7619

cc0 7620

caddc 7623

cpnf 7797

cmnf 7798

cxr 7799

cle 7801

cxad 9557

cicc 9674

cxmet 12149

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-xadd 9560 df-icc 9678 df-xmet 12157

This theorem is referenced by: (None)

W3C validator