Theorem ivthdec 14964

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 8423	. . 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 2196	. . . . 5 |- ( w e. D |-> -u ( F ` w ) ) = ( w e. D |-> -u ( F ` w ) )
9	8	negfcncf 14926	. . . 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 5561	. . . . . 6 |- ( w = x -> ( F ` w ) = ( F ` x ) )
12	11	negeqd 8238	. . . . 5 |- ( w = x -> -u ( F ` w ) = -u ( F ` x ) )
13	6	sselda 3184	. . . . 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 8423	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
16	8, 12, 13, 15	fvmptd3 5658	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
17	16, 15	eqeltrd 2273	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) e. RR )
18		fveq2 5561	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
19	18	negeqd 8238	. . . . . 6 |- ( w = A -> -u ( F ` w ) = -u ( F ` A ) )
20	1	rexrd 8093	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 8093	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 8162	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 10076	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	20, 21, 22, 23	syl3anc 1249	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
25	6, 24	sseldd 3185	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 5561	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
27	26	eleq1d 2265	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
28	14	ralrimiva 2570	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
29	27, 28, 24	rspcdva 2873	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
30	29	renegcld 8423	. . . . . 6 |- ( ph -> -u ( F ` A ) e. RR )
31	8, 19, 25, 30	fvmptd3 5658	. . . . 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 114	. . . . . 6 |- ( ph -> U < ( F ` A ) )
34	3, 29	ltnegd 8567	. . . . . 6 |- ( ph -> ( U < ( F ` A ) <-> -u ( F ` A ) < -u U ) )
35	33, 34	mpbid 147	. . . . 5 |- ( ph -> -u ( F ` A ) < -u U )
36	31, 35	eqbrtrd 4056	. . . 4 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U )
37	32	simpld 112	. . . . . 6 |- ( ph -> ( F ` B ) < U )
38		fveq2 5561	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39	38	eleq1d 2265	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
40		ubicc2 10077	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
41	20, 21, 22, 40	syl3anc 1249	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
42	39, 28, 41	rspcdva 2873	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
43	42, 3	ltnegd 8567	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
44	37, 43	mpbid 147	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
45		fveq2 5561	. . . . . . 7 |- ( w = B -> ( F ` w ) = ( F ` B ) )
46	45	negeqd 8238	. . . . . 6 |- ( w = B -> -u ( F ` w ) = -u ( F ` B ) )
47	6, 41	sseldd 3185	. . . . . 6 |- ( ph -> B e. D )
48	42	renegcld 8423	. . . . . 6 |- ( ph -> -u ( F ` B ) e. RR )
49	8, 46, 47, 48	fvmptd3 5658	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` B ) = -u ( F ` B ) )
50	44, 49	breqtrrd 4062	. . . 4 |- ( ph -> -u U < ( ( w e. D |-> -u ( F ` w ) ) ` B ) )
51	36, 50	jca 306	. . 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 5561	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
54	53	eleq1d 2265	. . . . . . 7 |- ( x = y -> ( ( F ` x ) e. RR <-> ( F ` y ) e. RR ) )
55		simpll 527	. . . . . . . 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 529	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> y e. ( A [,] B ) )
58	54, 56, 57	rspcdva 2873	. . . . . 6 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) e. RR )
59	14	adantr 276	. . . . . 6 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) e. RR )
60	58, 59	ltnegd 8567	. . . . 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 147	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` x ) < -u ( F ` y ) )
62	13	adantr 276	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> x e. D )
63	15	adantr 276	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` x ) e. RR )
64	8, 12, 62, 63	fvmptd3 5658	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
65		fveq2 5561	. . . . . 6 |- ( w = y -> ( F ` w ) = ( F ` y ) )
66	65	negeqd 8238	. . . . 5 |- ( w = y -> -u ( F ` w ) = -u ( F ` y ) )
67	6	sseld 3183	. . . . . 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 8423	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` y ) e. RR )
70	8, 66, 68, 69	fvmptd3 5658	. . . 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 4066	. . 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 14963	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U )
73		fveq2 5561	. . . . . . 7 |- ( w = c -> ( F ` w ) = ( F ` c ) )
74	73	negeqd 8238	. . . . . 6 |- ( w = c -> -u ( F ` w ) = -u ( F ` c ) )
75		ioossicc 10051	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
76	75, 6	sstrid 3195	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
77	76	sselda 3184	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
78		fveq2 5561	. . . . . . . . 9 |- ( x = c -> ( F ` x ) = ( F ` c ) )
79	78	eleq1d 2265	. . . . . . . 8 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
80	28	adantr 276	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
81	75	sseli 3180	. . . . . . . . 9 |- ( c e. ( A (,) B ) -> c e. ( A [,] B ) )
82	81	adantl 277	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. ( A [,] B ) )
83	79, 80, 82	rspcdva 2873	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
84	83	renegcld 8423	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> -u ( F ` c ) e. RR )
85	8, 74, 77, 84	fvmptd3 5658	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u ( F ` c ) )
86	85	eqeq1d 2205	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
87		cncff 14897	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
88	7, 87	syl 14	. . . . . . 7 |- ( ph -> F : D --> CC )
89	88	ffvelcdmda 5700	. . . . . 6 |- ( ( ph /\ c e. D ) -> ( F ` c ) e. CC )
90	77, 89	syldan 282	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. CC )
91	3	recnd 8072	. . . . . 6 |- ( ph -> U e. CC )
92	91	adantr 276	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
93	90, 92	neg11ad 8350	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( -u ( F ` c ) = -u U <-> ( F ` c ) = U ) )
94	86, 93	bitrd 188	. . 3 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> ( F ` c ) = U ) )
95	94	rexbidva 2494	. 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 147	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   class class class wbr 4034   |-> cmpt 4095   -->wf 5255   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   RR*cxr 8077   < clt 8078   <_ cle 8079   -ucneg 8215   (,)cioo 9980   [,]cicc 9983   -cn->ccncf 14890

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-pre-suploc 8017

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-ioo 9984  df-icc 9987  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-cncf 14891

This theorem is referenced by:  cosz12  15100  ioocosf1o  15174