Theorem ivthdec 14651

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 8378	. . 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 2189	. . . . 5 |- ( w e. D |-> -u ( F ` w ) ) = ( w e. D |-> -u ( F ` w ) )
9	8	negfcncf 14613	. . . 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 5541	. . . . . 6 |- ( w = x -> ( F ` w ) = ( F ` x ) )
12	11	negeqd 8193	. . . . 5 |- ( w = x -> -u ( F ` w ) = -u ( F ` x ) )
13	6	sselda 3174	. . . . 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 8378	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
16	8, 12, 13, 15	fvmptd3 5637	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
17	16, 15	eqeltrd 2266	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) e. RR )
18		fveq2 5541	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
19	18	negeqd 8193	. . . . . 6 |- ( w = A -> -u ( F ` w ) = -u ( F ` A ) )
20	1	rexrd 8048	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 8048	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 8117	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 10026	. . . . . . . 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 3175	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 5541	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
27	26	eleq1d 2258	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
28	14	ralrimiva 2563	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
29	27, 28, 24	rspcdva 2865	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
30	29	renegcld 8378	. . . . . 6 |- ( ph -> -u ( F ` A ) e. RR )
31	8, 19, 25, 30	fvmptd3 5637	. . . . 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 8521	. . . . . 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 4047	. . . 4 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U )
37	32	simpld 112	. . . . . 6 |- ( ph -> ( F ` B ) < U )
38		fveq2 5541	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39	38	eleq1d 2258	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
40		ubicc2 10027	. . . . . . . . 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 2865	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
43	42, 3	ltnegd 8521	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
44	37, 43	mpbid 147	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
45		fveq2 5541	. . . . . . 7 |- ( w = B -> ( F ` w ) = ( F ` B ) )
46	45	negeqd 8193	. . . . . 6 |- ( w = B -> -u ( F ` w ) = -u ( F ` B ) )
47	6, 41	sseldd 3175	. . . . . 6 |- ( ph -> B e. D )
48	42	renegcld 8378	. . . . . 6 |- ( ph -> -u ( F ` B ) e. RR )
49	8, 46, 47, 48	fvmptd3 5637	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` B ) = -u ( F ` B ) )
50	44, 49	breqtrrd 4053	. . . 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 5541	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
54	53	eleq1d 2258	. . . . . . 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 2865	. . . . . 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 8521	. . . . 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 5637	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
65		fveq2 5541	. . . . . 6 |- ( w = y -> ( F ` w ) = ( F ` y ) )
66	65	negeqd 8193	. . . . 5 |- ( w = y -> -u ( F ` w ) = -u ( F ` y ) )
67	6	sseld 3173	. . . . . 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 8378	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` y ) e. RR )
70	8, 66, 68, 69	fvmptd3 5637	. . . 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 4057	. . 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 14650	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U )
73		fveq2 5541	. . . . . . 7 |- ( w = c -> ( F ` w ) = ( F ` c ) )
74	73	negeqd 8193	. . . . . 6 |- ( w = c -> -u ( F ` w ) = -u ( F ` c ) )
75		ioossicc 10001	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
76	75, 6	sstrid 3185	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
77	76	sselda 3174	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
78		fveq2 5541	. . . . . . . . 9 |- ( x = c -> ( F ` x ) = ( F ` c ) )
79	78	eleq1d 2258	. . . . . . . 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 3170	. . . . . . . . 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 2865	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
84	83	renegcld 8378	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> -u ( F ` c ) e. RR )
85	8, 74, 77, 84	fvmptd3 5637	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u ( F ` c ) )
86	85	eqeq1d 2198	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
87		cncff 14585	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
88	7, 87	syl 14	. . . . . . 7 |- ( ph -> F : D --> CC )
89	88	ffvelcdmda 5679	. . . . . 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 8027	. . . . . 6 |- ( ph -> U e. CC )
92	91	adantr 276	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
93	90, 92	neg11ad 8305	. . . 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 2487	. 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 2160   A.wral 2468   E.wrex 2469   C_ wss 3148   class class class wbr 4025   |-> cmpt 4086   -->wf 5238   ` cfv 5242  (class class class)co 5904   CCcc 7849   RRcr 7850   RR*cxr 8032   < clt 8033   <_ cle 8034   -ucneg 8170   (,)cioo 9930   [,]cicc 9933   -cn->ccncf 14578

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7942  ax-resscn 7943  ax-1cn 7944  ax-1re 7945  ax-icn 7946  ax-addcl 7947  ax-addrcl 7948  ax-mulcl 7949  ax-mulrcl 7950  ax-addcom 7951  ax-mulcom 7952  ax-addass 7953  ax-mulass 7954  ax-distr 7955  ax-i2m1 7956  ax-0lt1 7957  ax-1rid 7958  ax-0id 7959  ax-rnegex 7960  ax-precex 7961  ax-cnre 7962  ax-pre-ltirr 7963  ax-pre-ltwlin 7964  ax-pre-lttrn 7965  ax-pre-apti 7966  ax-pre-ltadd 7967  ax-pre-mulgt0 7968  ax-pre-mulext 7969  ax-arch 7970  ax-caucvg 7971  ax-pre-suploc 7972

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-isom 5251  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-frec 6424  df-map 6684  df-sup 7022  df-inf 7023  df-pnf 8035  df-mnf 8036  df-xr 8037  df-ltxr 8038  df-le 8039  df-sub 8171  df-neg 8172  df-reap 8573  df-ap 8580  df-div 8671  df-inn 8961  df-2 9019  df-3 9020  df-4 9021  df-n0 9218  df-z 9295  df-uz 9570  df-rp 9696  df-ioo 9934  df-icc 9937  df-seqfrec 10491  df-exp 10566  df-cj 10898  df-re 10899  df-im 10900  df-rsqrt 11054  df-abs 11055  df-cncf 14579

This theorem is referenced by:  cosz12  14739  ioocosf1o  14813