Theorem ivthdec 13501

Description: The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)

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 )
ivthdec.9	|- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )
ivthdec.i	|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) < ( F ` x ) )

Assertion

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

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

Proof of Theorem ivthdec

Dummy variable w 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 8303	. . 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 2171	. . . . 5 |- ( w e. D |-> -u ( F ` w ) ) = ( w e. D |-> -u ( F ` w ) )
9	8	negfcncf 13468	. . . 4 |- ( F e. ( D -cn-> CC ) -> ( w e. D |-> -u ( F ` w ) ) e. ( D -cn-> CC ) )
10	7, 9	syl 14	. . 3 |- ( ph -> ( w e. D |-> -u ( F ` w ) ) e. ( D -cn-> CC ) )
11		fveq2 5499	. . . . . 6 |- ( w = x -> ( F ` w ) = ( F ` x ) )
12	11	negeqd 8118	. . . . 5 |- ( w = x -> -u ( F ` w ) = -u ( F ` x ) )
13	6	sselda 3148	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. D )
14		ivth.8	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
15	14	renegcld 8303	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
16	8, 12, 13, 15	fvmptd3 5593	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
17	16, 15	eqeltrd 2248	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) e. RR )
18		fveq2 5499	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
19	18	negeqd 8118	. . . . . 6 |- ( w = A -> -u ( F ` w ) = -u ( F ` A ) )
20	1	rexrd 7973	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 7973	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 8042	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 9945	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	20, 21, 22, 23	syl3anc 1234	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
25	6, 24	sseldd 3149	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 5499	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
27	26	eleq1d 2240	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
28	14	ralrimiva 2544	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
29	27, 28, 24	rspcdva 2840	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
30	29	renegcld 8303	. . . . . 6 |- ( ph -> -u ( F ` A ) e. RR )
31	8, 19, 25, 30	fvmptd3 5593	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) = -u ( F ` A ) )
32		ivthdec.9	. . . . . . 7 |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )
33	32	simprd 113	. . . . . 6 |- ( ph -> U < ( F ` A ) )
34	3, 29	ltnegd 8446	. . . . . 6 |- ( ph -> ( U < ( F ` A ) <-> -u ( F ` A ) < -u U ) )
35	33, 34	mpbid 146	. . . . 5 |- ( ph -> -u ( F ` A ) < -u U )
36	31, 35	eqbrtrd 4012	. . . 4 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U )
37	32	simpld 111	. . . . . 6 |- ( ph -> ( F ` B ) < U )
38		fveq2 5499	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39	38	eleq1d 2240	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
40		ubicc2 9946	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
41	20, 21, 22, 40	syl3anc 1234	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
42	39, 28, 41	rspcdva 2840	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
43	42, 3	ltnegd 8446	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
44	37, 43	mpbid 146	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
45		fveq2 5499	. . . . . . 7 |- ( w = B -> ( F ` w ) = ( F ` B ) )
46	45	negeqd 8118	. . . . . 6 |- ( w = B -> -u ( F ` w ) = -u ( F ` B ) )
47	6, 41	sseldd 3149	. . . . . 6 |- ( ph -> B e. D )
48	42	renegcld 8303	. . . . . 6 |- ( ph -> -u ( F ` B ) e. RR )
49	8, 46, 47, 48	fvmptd3 5593	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` B ) = -u ( F ` B ) )
50	44, 49	breqtrrd 4018	. . . 4 |- ( ph -> -u U < ( ( w e. D |-> -u ( F ` w ) ) ` B ) )
51	36, 50	jca 304	. . 3 |- ( ph -> ( ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U /\ -u U < ( ( w e. D |-> -u ( F ` w ) ) ` B ) ) )
52		ivthdec.i	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) < ( F ` x ) )
53		fveq2 5499	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
54	53	eleq1d 2240	. . . . . . 7 |- ( x = y -> ( ( F ` x ) e. RR <-> ( F ` y ) e. RR ) )
55		simpll 525	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ph )
56	55, 28	syl 14	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
57		simprl 527	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> y e. ( A [,] B ) )
58	54, 56, 57	rspcdva 2840	. . . . . 6 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) e. RR )
59	14	adantr 274	. . . . . 6 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) e. RR )
60	58, 59	ltnegd 8446	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( F ` y ) < ( F ` x ) <-> -u ( F ` x ) < -u ( F ` y ) ) )
61	52, 60	mpbid 146	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` x ) < -u ( F ` y ) )
62	13	adantr 274	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> x e. D )
63	15	adantr 274	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` x ) e. RR )
64	8, 12, 62, 63	fvmptd3 5593	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
65		fveq2 5499	. . . . . 6 |- ( w = y -> ( F ` w ) = ( F ` y ) )
66	65	negeqd 8118	. . . . 5 |- ( w = y -> -u ( F ` w ) = -u ( F ` y ) )
67	6	sseld 3147	. . . . . 6 |- ( ph -> ( y e. ( A [,] B ) -> y e. D ) )
68	55, 57, 67	sylc 62	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> y e. D )
69	58	renegcld 8303	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` y ) e. RR )
70	8, 66, 68, 69	fvmptd3 5593	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` y ) = -u ( F ` y ) )
71	61, 64, 70	3brtr4d 4022	. . 3 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) < ( ( w e. D |-> -u ( F ` w ) ) ` y ) )
72	1, 2, 4, 5, 6, 10, 17, 51, 71	ivthinc 13500	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U )
73		fveq2 5499	. . . . . . 7 |- ( w = c -> ( F ` w ) = ( F ` c ) )
74	73	negeqd 8118	. . . . . 6 |- ( w = c -> -u ( F ` w ) = -u ( F ` c ) )
75		ioossicc 9920	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
76	75, 6	sstrid 3159	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
77	76	sselda 3148	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
78		fveq2 5499	. . . . . . . . 9 |- ( x = c -> ( F ` x ) = ( F ` c ) )
79	78	eleq1d 2240	. . . . . . . 8 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
80	28	adantr 274	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
81	75	sseli 3144	. . . . . . . . 9 |- ( c e. ( A (,) B ) -> c e. ( A [,] B ) )
82	81	adantl 275	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. ( A [,] B ) )
83	79, 80, 82	rspcdva 2840	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
84	83	renegcld 8303	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> -u ( F ` c ) e. RR )
85	8, 74, 77, 84	fvmptd3 5593	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u ( F ` c ) )
86	85	eqeq1d 2180	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
87		cncff 13443	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
88	7, 87	syl 14	. . . . . . 7 |- ( ph -> F : D --> CC )
89	88	ffvelcdmda 5635	. . . . . 6 |- ( ( ph /\ c e. D ) -> ( F ` c ) e. CC )
90	77, 89	syldan 280	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. CC )
91	3	recnd 7952	. . . . . 6 |- ( ph -> U e. CC )
92	91	adantr 274	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
93	90, 92	neg11ad 8230	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( -u ( F ` c ) = -u U <-> ( F ` c ) = U ) )
94	86, 93	bitrd 187	. . 3 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> ( F ` c ) = U ) )
95	94	rexbidva 2468	. 2 |- ( ph -> ( E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> E. c e. ( A (,) B ) ( F ` c ) = U ) )
96	72, 95	mpbid 146	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1349   e. wcel 2142   A.wral 2449   E.wrex 2450   C_ wss 3122   class class class wbr 3990   |-> cmpt 4051   -->wf 5196   ` cfv 5200  (class class class)co 5857   CCcc 7776   RRcr 7777   RR*cxr 7957   < clt 7958   <_ cle 7959   -ucneg 8095   (,)cioo 9849   [,]cicc 9852   -cn->ccncf 13436

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895  ax-pre-mulext 7896  ax-arch 7897  ax-caucvg 7898  ax-pre-suploc 7899

This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-isom 5209  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-map 6632  df-sup 6965  df-inf 6966  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-reap 8498  df-ap 8505  df-div 8594  df-inn 8883  df-2 8941  df-3 8942  df-4 8943  df-n0 9140  df-z 9217  df-uz 9492  df-rp 9615  df-ioo 9853  df-icc 9856  df-seqfrec 10406  df-exp 10480  df-cj 10810  df-re 10811  df-im 10812  df-rsqrt 10966  df-abs 10967  df-cncf 13437

This theorem is referenced by:  cosz12  13580  ioocosf1o  13654