Theorem ivthdec 13262

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 8278	. . 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 2165	. . . . 5 |- ( w e. D |-> -u ( F ` w ) ) = ( w e. D |-> -u ( F ` w ) )
9	8	negfcncf 13229	. . . 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 5486	. . . . . 6 |- ( w = x -> ( F ` w ) = ( F ` x ) )
12	11	negeqd 8093	. . . . 5 |- ( w = x -> -u ( F ` w ) = -u ( F ` x ) )
13	6	sselda 3142	. . . . 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 8278	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u ( F ` x ) e. RR )
16	8, 12, 13, 15	fvmptd3 5579	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
17	16, 15	eqeltrd 2243	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) e. RR )
18		fveq2 5486	. . . . . . 7 |- ( w = A -> ( F ` w ) = ( F ` A ) )
19	18	negeqd 8093	. . . . . 6 |- ( w = A -> -u ( F ` w ) = -u ( F ` A ) )
20	1	rexrd 7948	. . . . . . . 8 |- ( ph -> A e. RR* )
21	2	rexrd 7948	. . . . . . . 8 |- ( ph -> B e. RR* )
22	1, 2, 5	ltled 8017	. . . . . . . 8 |- ( ph -> A <_ B )
23		lbicc2 9920	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
24	20, 21, 22, 23	syl3anc 1228	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
25	6, 24	sseldd 3143	. . . . . 6 |- ( ph -> A e. D )
26		fveq2 5486	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
27	26	eleq1d 2235	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) e. RR <-> ( F ` A ) e. RR ) )
28	14	ralrimiva 2539	. . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
29	27, 28, 24	rspcdva 2835	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
30	29	renegcld 8278	. . . . . 6 |- ( ph -> -u ( F ` A ) e. RR )
31	8, 19, 25, 30	fvmptd3 5579	. . . . 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 8421	. . . . . 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 4004	. . . 4 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` A ) < -u U )
37	32	simpld 111	. . . . . 6 |- ( ph -> ( F ` B ) < U )
38		fveq2 5486	. . . . . . . . 9 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39	38	eleq1d 2235	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) e. RR <-> ( F ` B ) e. RR ) )
40		ubicc2 9921	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
41	20, 21, 22, 40	syl3anc 1228	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
42	39, 28, 41	rspcdva 2835	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
43	42, 3	ltnegd 8421	. . . . . 6 |- ( ph -> ( ( F ` B ) < U <-> -u U < -u ( F ` B ) ) )
44	37, 43	mpbid 146	. . . . 5 |- ( ph -> -u U < -u ( F ` B ) )
45		fveq2 5486	. . . . . . 7 |- ( w = B -> ( F ` w ) = ( F ` B ) )
46	45	negeqd 8093	. . . . . 6 |- ( w = B -> -u ( F ` w ) = -u ( F ` B ) )
47	6, 41	sseldd 3143	. . . . . 6 |- ( ph -> B e. D )
48	42	renegcld 8278	. . . . . 6 |- ( ph -> -u ( F ` B ) e. RR )
49	8, 46, 47, 48	fvmptd3 5579	. . . . 5 |- ( ph -> ( ( w e. D |-> -u ( F ` w ) ) ` B ) = -u ( F ` B ) )
50	44, 49	breqtrrd 4010	. . . 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 5486	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
54	53	eleq1d 2235	. . . . . . 7 |- ( x = y -> ( ( F ` x ) e. RR <-> ( F ` y ) e. RR ) )
55		simpll 519	. . . . . . . 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 521	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> y e. ( A [,] B ) )
58	54, 56, 57	rspcdva 2835	. . . . . 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 8421	. . . . 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 5579	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` x ) = -u ( F ` x ) )
65		fveq2 5486	. . . . . 6 |- ( w = y -> ( F ` w ) = ( F ` y ) )
66	65	negeqd 8093	. . . . 5 |- ( w = y -> -u ( F ` w ) = -u ( F ` y ) )
67	6	sseld 3141	. . . . . 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 8278	. . . . 5 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> -u ( F ` y ) e. RR )
70	8, 66, 68, 69	fvmptd3 5579	. . . 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 4014	. . 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 13261	. 2 |- ( ph -> E. c e. ( A (,) B ) ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U )
73		fveq2 5486	. . . . . . 7 |- ( w = c -> ( F ` w ) = ( F ` c ) )
74	73	negeqd 8093	. . . . . 6 |- ( w = c -> -u ( F ` w ) = -u ( F ` c ) )
75		ioossicc 9895	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
76	75, 6	sstrid 3153	. . . . . . 7 |- ( ph -> ( A (,) B ) C_ D )
77	76	sselda 3142	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. D )
78		fveq2 5486	. . . . . . . . 9 |- ( x = c -> ( F ` x ) = ( F ` c ) )
79	78	eleq1d 2235	. . . . . . . 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 3138	. . . . . . . . 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 2835	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
84	83	renegcld 8278	. . . . . 6 |- ( ( ph /\ c e. ( A (,) B ) ) -> -u ( F ` c ) e. RR )
85	8, 74, 77, 84	fvmptd3 5579	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u ( F ` c ) )
86	85	eqeq1d 2174	. . . 4 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( ( w e. D |-> -u ( F ` w ) ) ` c ) = -u U <-> -u ( F ` c ) = -u U ) )
87		cncff 13204	. . . . . . . 8 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
88	7, 87	syl 14	. . . . . . 7 |- ( ph -> F : D --> CC )
89	88	ffvelrnda 5620	. . . . . 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 7927	. . . . . 6 |- ( ph -> U e. CC )
92	91	adantr 274	. . . . 5 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. CC )
93	90, 92	neg11ad 8205	. . . 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 2463	. 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 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   C_ wss 3116   class class class wbr 3982   |-> cmpt 4043   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   RR*cxr 7932   < clt 7933   <_ cle 7934   -ucneg 8070   (,)cioo 9824   [,]cicc 9827   -cn->ccncf 13197

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-ioo 9828  df-icc 9831  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-cncf 13198

This theorem is referenced by:  cosz12  13341  ioocosf1o  13415