Theorem ivthdec 15509

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 8653	. . 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 2232	. . . . 5 |- ( w e. D |-> -u ( F ` w ) ) = ( w e. D |-> -u ( F ` w ) )
9	8	negfcncf 15471	. . . 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 5670	. . . . . 6 |- ( w = x -> ( F ` w ) = ( F ` x ) )
12	11	negeqd 8468	. . . . 5 |- ( w = x -> -u ( F ` w ) = -u ( F ` x ) )
13	6	sselda 3238	. . . . 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 8653	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
16	8, 12, 13, 15	fvmptd3 5771	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
17	16, 15	eqeltrd 2309	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) e. RR )
18		fveq2 5670	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
19	18	negeqd 8468	. . . . . 6 |- ( w = A -> -u ( F ` w ) = -u ( F ` A ) )
20	1	rexrd 8323	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 8323	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 8392	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 10317	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	20, 21, 22, 23	syl3anc 1274	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
25	6, 24	sseldd 3239	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 5670	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
27	26	eleq1d 2301	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
28	14	ralrimiva 2615	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
29	27, 28, 24	rspcdva 2926	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
30	29	renegcld 8653	. . . . . 6 |- ( ph -> -u ( F ` A ) e. RR )
31	8, 19, 25, 30	fvmptd3 5771	. . . . 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 8797	. . . . . 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 4131	. . . 4 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U )
37	32	simpld 112	. . . . . 6 |- ( ph -> ( F ` B ) < U )
38		fveq2 5670	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39	38	eleq1d 2301	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
40		ubicc2 10318	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
41	20, 21, 22, 40	syl3anc 1274	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
42	39, 28, 41	rspcdva 2926	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
43	42, 3	ltnegd 8797	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
44	37, 43	mpbid 147	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
45		fveq2 5670	. . . . . . 7 |- ( w = B -> ( F ` w ) = ( F ` B ) )
46	45	negeqd 8468	. . . . . 6 |- ( w = B -> -u ( F ` w ) = -u ( F ` B ) )
47	6, 41	sseldd 3239	. . . . . 6 |- ( ph -> B e. D )
48	42	renegcld 8653	. . . . . 6 |- ( ph -> -u ( F ` B ) e. RR )
49	8, 46, 47, 48	fvmptd3 5771	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` B ) = -u ( F ` B ) )
50	44, 49	breqtrrd 4137	. . . 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 5670	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
54	53	eleq1d 2301	. . . . . . 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 531	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> y e. ( A [,] B ) )
58	54, 56, 57	rspcdva 2926	. . . . . 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 8797	. . . . 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 5771	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
65		fveq2 5670	. . . . . 6 |- ( w = y -> ( F ` w ) = ( F ` y ) )
66	65	negeqd 8468	. . . . 5 |- ( w = y -> -u ( F ` w ) = -u ( F ` y ) )
67	6	sseld 3237	. . . . . 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 8653	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` y ) e. RR )
70	8, 66, 68, 69	fvmptd3 5771	. . . 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 4141	. . 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 15508	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U )
73		fveq2 5670	. . . . . . 7 |- ( w = c -> ( F ` w ) = ( F ` c ) )
74	73	negeqd 8468	. . . . . 6 |- ( w = c -> -u ( F ` w ) = -u ( F ` c ) )
75		ioossicc 10292	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
76	75, 6	sstrid 3249	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
77	76	sselda 3238	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
78		fveq2 5670	. . . . . . . . 9 |- ( x = c -> ( F ` x ) = ( F ` c ) )
79	78	eleq1d 2301	. . . . . . . 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 3234	. . . . . . . . 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 2926	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
84	83	renegcld 8653	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> -u ( F ` c ) e. RR )
85	8, 74, 77, 84	fvmptd3 5771	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u ( F ` c ) )
86	85	eqeq1d 2241	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
87		cncff 15442	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
88	7, 87	syl 14	. . . . . . 7 |- ( ph -> F : D --> CC )
89	88	ffvelcdmda 5812	. . . . . 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 8302	. . . . . 6 |- ( ph -> U e. CC )
92	91	adantr 276	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
93	90, 92	neg11ad 8580	. . . 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 2539	. 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 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   C_ wss 3211   class class class wbr 4109   |-> cmpt 4171   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   RR*cxr 8307   < clt 8308   <_ cle 8309   -ucneg 8445   (,)cioo 10221   [,]cicc 10224   -cn->ccncf 15435

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247  ax-pre-suploc 8248

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-ioo 10225  df-icc 10228  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-cncf 15436

This theorem is referenced by:  cosz12  15645  ioocosf1o  15719