pmltpc - Metamath Proof Explorer

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Theorem pmltpc 23219

Description: Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)

**Assertion**
Ref	Expression
pmltpc	$/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$

Distinct variable groups:

Proof of Theorem pmltpc Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexanali 2998	. . . . . . . 8 $/\$
2	1	rexbii 3041	. . . . . . 7 $/\$
3		rexnal 2995	. . . . . . 7
4	2, 3	bitri 264	. . . . . 6 $/\$
5		rexanali 2998	. . . . . . . 8 $/\$
6	5	rexbii 3041	. . . . . . 7 $/\$
7		rexnal 2995	. . . . . . . 8
8		breq1 4656	. . . . . . . . . 10
9		fveq2 6191	. . . . . . . . . . 11
10	9	breq2d 4665	. . . . . . . . . 10
11	8, 10	imbi12d 334	. . . . . . . . 9
12		breq2 4657	. . . . . . . . . 10
13		fveq2 6191	. . . . . . . . . . 11
14	13	breq1d 4663	. . . . . . . . . 10
15	12, 14	imbi12d 334	. . . . . . . . 9
16	11, 15	cbvral2v 3179	. . . . . . . 8
17	7, 16	xchbinx 324	. . . . . . 7
18	6, 17	bitri 264	. . . . . 6 $/\$
19	4, 18	anbi12i 733	. . . . 5 $/\$ $/\$ $/\$ $/\$
20		reeanv 3107	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		ioran 511	. . . . 5 $\/$ $/\$
22	19, 20, 21	3bitr4i 292	. . . 4 $/\$ $/\$ $/\$ $\/$
23		reeanv 3107	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		simplll 798	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	24	simpld 475	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	24	simprd 479	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpllr 799	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	27	simpld 475	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simplrl 800	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	27	simprd 479	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrr 801	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simprll 802	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		simprrl 804	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		simprlr 803	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simprrr 805	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 26, 28, 29, 30, 31, 32, 33, 34, 35	pmltpclem2 23218	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
37	36	ex 450	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
38	37	rexlimdvva 3038	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
39	23, 38	syl5bir 233	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
40	39	rexlimdvva 3038	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
41	22, 40	syl5bir 233	. . 3 $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$
42	41	orrd 393	. 2 $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$
43		df-3or 1038	. 2 $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$
44	42, 43	sylibr 224	1 $/\$ $\/$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036 $/\$ w3a 1037

wcel 1990

wral 2912

wrex 2913

wss 3574 class class class wbr 4653

cdm 5114

cfv 5888 (class class class)co 6650

cpm 7858

cr 9935

clt 10074

cle 10075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080

This theorem is referenced by: (None)

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