isxmet2d - Intuitionistic Logic Explorer

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

Theorem isxmet2d 14820

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 6090	. . . 4 $/\$ $/\$
4		0xr 8119	. . . 4
5		xrletri3 9926	. . . 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 1206	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
13		rexadd 9974	. . . . . . 7 $/\$
14	13	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	12, 14	breqtrrd 4072	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	anassrs 400	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
17	3	3adantr3 1161	. . . . . . 7 $/\$ $/\$ $/\$
18		pnfge 9911	. . . . . . 7
19	17, 18	syl 14	. . . . . 6 $/\$ $/\$ $/\$
20	19	ad2antrr 488	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
21		oveq2 5952	. . . . . 6
22	2	ffnd 5426	. . . . . . . . . . 11
23		elxrge0 10100	. . . . . . . . . . . . 13 $/\$
24	3, 7, 23	sylanbrc 417	. . . . . . . . . . . 12 $/\$ $/\$
25	24	ralrimivva 2588	. . . . . . . . . . 11
26		ffnov 6049	. . . . . . . . . . 11 $/\$
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10
28	27	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
29		simpr3 1008	. . . . . . . . 9 $/\$ $/\$ $/\$
30		simpr1 1006	. . . . . . . . 9 $/\$ $/\$ $/\$
31	28, 29, 30	fovcdmd 6091	. . . . . . . 8 $/\$ $/\$ $/\$
32		elxrge0 10100	. . . . . . . . 9 $/\$
33	32	simplbi 274	. . . . . . . 8
34	31, 33	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
35		renemnf 8121	. . . . . . 7
36		xaddpnf1 9968	. . . . . . 7 $/\$
37	34, 35, 36	syl2an 289	. . . . . 6 $/\$ $/\$ $/\$ $/\$
38	21, 37	sylan9eqr 2260	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
39	20, 38	breqtrrd 4072	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 1007	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	28, 29, 40	fovcdmd 6091	. . . . . . . . . 10 $/\$ $/\$ $/\$
42		elxrge0 10100	. . . . . . . . . . 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 9946	. . . . . . . . 9 $/\$
48	44, 46, 47	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
49	48	neneqd 2397	. . . . . . 7 $/\$ $/\$ $/\$
50	49	pm2.21d 620	. . . . . 6 $/\$ $/\$ $/\$
51	50	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
52	51	imp 124	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	44	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
54		elxr 9898	. . . . 5 $\/$ $\/$
55	53, 54	sylib 122	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
56	16, 39, 52, 55	mpjao3dan 1320	. . 3 $/\$ $/\$ $/\$ $/\$
57	19	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
58		oveq1 5951	. . . . 5
59		xaddpnf2 9969	. . . . . 6 $/\$
60	44, 48, 59	syl2anc 411	. . . . 5 $/\$ $/\$ $/\$
61	58, 60	sylan9eqr 2260	. . . 4 $/\$ $/\$ $/\$ $/\$
62	57, 61	breqtrrd 4072	. . 3 $/\$ $/\$ $/\$ $/\$
63	32	simprbi 275	. . . . . . . 8
64	31, 63	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
65		ge0nemnf 9946	. . . . . . 7 $/\$
66	34, 64, 65	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
67	66	neneqd 2397	. . . . 5 $/\$ $/\$ $/\$
68	67	pm2.21d 620	. . . 4 $/\$ $/\$ $/\$
69	68	imp 124	. . 3 $/\$ $/\$ $/\$ $/\$
70		elxr 9898	. . . 4 $\/$ $\/$
71	34, 70	sylib 122	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
72	56, 62, 69, 71	mpjao3dan 1320	. 2 $/\$ $/\$ $/\$
73	1, 2, 10, 72	isxmetd 14819	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ w3o 980 $/\$ w3a 981

wceq 1373

wcel 2176

wne 2376

wral 2484

cvv 2772 class class class wbr 4044

cxp 4673

wfn 5266

wf 5267

cfv 5271 (class class class)co 5944

cr 7924

cc0 7925

caddc 7928

cpnf 8104

cmnf 8105

cxr 8106

cle 8108

cxad 9892

cicc 10013

cxmet 14298

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1re 8019 ax-addrcl 8022 ax-rnegex 8034 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-po 4343 df-iso 4344 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-map 6737 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-xadd 9895 df-icc 10017 df-xmet 14306

This theorem is referenced by: (None)

W3C validator