Theorem ivth2 23224

Description: The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Hypotheses

Ref	Expression
ivth.1	|- ( ph -> A e. RR )
ivth.2	|- ( ph -> B e. RR )
ivth.3	|- ( ph -> U e. RR )
ivth.4	|- ( ph -> A < B )
ivth.5	|- ( ph -> ( A [,] B ) C_ D )
ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivth2.9	|- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )

Assertion

Ref	Expression
ivth2	|- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Distinct variable groups:   x, c, B   D, c, x   F, c, x   ph, c, x   A, c, x   U, c, x

Proof of Theorem ivth2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivth.1	. . 3 |- ( ph -> A e. RR )
2		ivth.2	. . 3 |- ( ph -> B e. RR )
3		ivth.3	. . . 4 |- ( ph -> U e. RR )
4	3	renegcld 10457	. . 3 |- ( ph -> -u U e. RR )
5		ivth.4	. . 3 |- ( ph -> A < B )
6		ivth.5	. . 3 |- ( ph -> ( A [,] B ) C_ D )
7		ivth.7	. . . 4 |- ( ph -> F e. ( D -cn-> CC ) )
8		eqid 2622	. . . . 5 |- ( y e. D |-> -u ( F ` y ) ) = ( y e. D |-> -u ( F ` y ) )
9	8	negfcncf 22722	. . . 4 |- ( F e. ( D -cn-> CC ) -> ( y e. D |-> -u ( F ` y ) ) e. ( D -cn-> CC ) )
10	7, 9	syl 17	. . 3 |- ( ph -> ( y e. D |-> -u ( F ` y ) ) e. ( D -cn-> CC ) )
11	6	sselda 3603	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. D )
12		fveq2 6191	. . . . . . 7 |- ( y = x -> ( F ` y ) = ( F ` x ) )
13	12	negeqd 10275	. . . . . 6 |- ( y = x -> -u ( F ` y ) = -u ( F ` x ) )
14		negex 10279	. . . . . 6 |- -u ( F ` x ) e. _V
15	13, 8, 14	fvmpt 6282	. . . . 5 |- ( x e. D -> ( ( y e. D |-> -u ( F ` y ) ) ` x ) = -u ( F ` x ) )
16	11, 15	syl 17	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( y e. D |-> -u ( F ` y ) ) ` x ) = -u ( F ` x ) )
17		ivth.8	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
18	17	renegcld 10457	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
19	16, 18	eqeltrd 2701	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( y e. D |-> -u ( F ` y ) ) ` x ) e. RR )
20	1	rexrd 10089	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 10089	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 10185	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 12288	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	20, 21, 22, 23	syl3anc 1326	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
25	6, 24	sseldd 3604	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 6191	. . . . . . . 8 |- ( y = A -> ( F ` y ) = ( F ` A ) )
27	26	negeqd 10275	. . . . . . 7 |- ( y = A -> -u ( F ` y ) = -u ( F ` A ) )
28		negex 10279	. . . . . . 7 |- -u ( F ` A ) e. _V
29	27, 8, 28	fvmpt 6282	. . . . . 6 |- ( A e. D -> ( ( y e. D |-> -u ( F ` y ) ) ` A ) = -u ( F ` A ) )
30	25, 29	syl 17	. . . . 5 |- ( ph -> ( ( y e. D |-> -u ( F ` y ) ) ` A ) = -u ( F ` A ) )
31		ivth2.9	. . . . . . 7 |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )
32	31	simprd 479	. . . . . 6 |- ( ph -> U < ( F ` A ) )
33	17	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
34		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
35	34	eleq1d 2686	. . . . . . . . 9 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
36	35	rspcv 3305	. . . . . . . 8 |- ( A e. ( A [,] B ) -> ( A. x e. ( A [,] B ) ( F ` x ) e. RR -> ( F ` A ) e. RR ) )
37	24, 33, 36	sylc 65	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
38	3, 37	ltnegd 10605	. . . . . 6 |- ( ph -> ( U < ( F ` A ) <-> -u ( F ` A ) < -u U ) )
39	32, 38	mpbid 222	. . . . 5 |- ( ph -> -u ( F ` A ) < -u U )
40	30, 39	eqbrtrd 4675	. . . 4 |- ( ph -> ( ( y e. D |-> -u ( F ` y ) ) ` A ) < -u U )
41	31	simpld 475	. . . . . 6 |- ( ph -> ( F ` B ) < U )
42		ubicc2 12289	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
43	20, 21, 22, 42	syl3anc 1326	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
44		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
45	44	eleq1d 2686	. . . . . . . . 9 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
46	45	rspcv 3305	. . . . . . . 8 |- ( B e. ( A [,] B ) -> ( A. x e. ( A [,] B ) ( F ` x ) e. RR -> ( F ` B ) e. RR ) )
47	43, 33, 46	sylc 65	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
48	47, 3	ltnegd 10605	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
49	41, 48	mpbid 222	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
50	6, 43	sseldd 3604	. . . . . 6 |- ( ph -> B e. D )
51		fveq2 6191	. . . . . . . 8 |- ( y = B -> ( F ` y ) = ( F ` B ) )
52	51	negeqd 10275	. . . . . . 7 |- ( y = B -> -u ( F ` y ) = -u ( F ` B ) )
53		negex 10279	. . . . . . 7 |- -u ( F ` B ) e. _V
54	52, 8, 53	fvmpt 6282	. . . . . 6 |- ( B e. D -> ( ( y e. D |-> -u ( F ` y ) ) ` B ) = -u ( F ` B ) )
55	50, 54	syl 17	. . . . 5 |- ( ph -> ( ( y e. D |-> -u ( F ` y ) ) ` B ) = -u ( F ` B ) )
56	49, 55	breqtrrd 4681	. . . 4 |- ( ph -> -u U < ( ( y e. D |-> -u ( F ` y ) ) ` B ) )
57	40, 56	jca 554	. . 3 |- ( ph -> ( ( ( y e. D |-> -u ( F ` y ) ) ` A ) < -u U /\ -u U < ( ( y e. D |-> -u ( F ` y ) ) ` B ) ) )
58	1, 2, 4, 5, 6, 10, 19, 57	ivth 23223	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u U )
59		ioossicc 12259	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
60	59, 6	syl5ss 3614	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
61	60	sselda 3603	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
62		fveq2 6191	. . . . . . . 8 |- ( y = c -> ( F ` y ) = ( F ` c ) )
63	62	negeqd 10275	. . . . . . 7 |- ( y = c -> -u ( F ` y ) = -u ( F ` c ) )
64		negex 10279	. . . . . . 7 |- -u ( F ` c ) e. _V
65	63, 8, 64	fvmpt 6282	. . . . . 6 |- ( c e. D -> ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u ( F ` c ) )
66	61, 65	syl 17	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u ( F ` c ) )
67	66	eqeq1d 2624	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
68		cncff 22696	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
69	7, 68	syl 17	. . . . . . 7 |- ( ph -> F : D --> CC )
70	69	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ c e. D ) -> ( F ` c ) e. CC )
71	61, 70	syldan 487	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. CC )
72	3	recnd 10068	. . . . . 6 |- ( ph -> U e. CC )
73	72	adantr 481	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
74	71, 73	neg11ad 10388	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( -u ( F ` c ) = -u U <-> ( F ` c ) = U ) )
75	67, 74	bitrd 268	. . 3 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u U <-> ( F ` c ) = U ) )
76	75	rexbidva 3049	. 2 |- ( ph -> ( E. c e. ( A (,) B ) ( ( y e. D |-> -u ( F ` y ) ) ` c ) = -u U <-> E. c e. ( A (,) B ) ( F ` c ) = U ) )
77	58, 76	mpbid 222	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681

This theorem is referenced by:  ivthle2  23226  pilem3  24207  signsply0  30628