Theorem ivthinc 13415

Description: The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-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 )
ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivthinc.i	|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )

Assertion

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

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

Allowed substitution hints:   D( x, y, c)

Proof of Theorem ivthinc

Dummy variables p r w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivth.1	. . . 4 |- ( ph -> A e. RR )
2		ivth.2	. . . 4 |- ( ph -> B e. RR )
3		ivth.3	. . . 4 |- ( ph -> U e. RR )
4		ivth.4	. . . 4 |- ( ph -> A < B )
5		ivth.5	. . . 4 |- ( ph -> ( A [,] B ) C_ D )
6		ivth.7	. . . 4 |- ( ph -> F e. ( D -cn-> CC ) )
7		ivth.8	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
8		ivth.9	. . . 4 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
9		ivthinc.i	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
10		eqid 2170	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } = { w e. ( A [,] B ) | ( F ` w ) < U }
11		eqid 2170	. . . 4 |- { w e. ( A [,] B ) | U < ( F ` w ) } = { w e. ( A [,] B ) | U < ( F ` w ) }
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	ivthinclemex 13414	. . 3 |- ( ph -> E! c e. ( A (,) B ) ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) )
13		reurex 2683	. . 3 |- ( E! c e. ( A (,) B ) ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) -> E. c e. ( A (,) B ) ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) )
14	12, 13	syl 14	. 2 |- ( ph -> E. c e. ( A (,) B ) ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) )
15		elioore 9869	. . . . . . . . . 10 |- ( c e. ( A (,) B ) -> c e. RR )
16	15	ad2antlr 486	. . . . . . . . 9 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> c e. RR )
17	16	ltnrd 8031	. . . . . . . 8 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> -. c < c )
18		breq1 3992	. . . . . . . . 9 |- ( p = c -> ( p < c <-> c < c ) )
19		simplrl 530	. . . . . . . . 9 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ ( F ` c ) < U ) -> A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c )
20		fveq2 5496	. . . . . . . . . . 11 |- ( w = c -> ( F ` w ) = ( F ` c ) )
21	20	breq1d 3999	. . . . . . . . . 10 |- ( w = c -> ( ( F ` w ) < U <-> ( F ` c ) < U ) )
22		ioossicc 9916	. . . . . . . . . . . . 13 |- ( A (,) B ) C_ ( A [,] B )
23	22	sseli 3143	. . . . . . . . . . . 12 |- ( c e. ( A (,) B ) -> c e. ( A [,] B ) )
24	23	adantl 275	. . . . . . . . . . 11 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. ( A [,] B ) )
25	24	ad2antrr 485	. . . . . . . . . 10 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ ( F ` c ) < U ) -> c e. ( A [,] B ) )
26		simpr 109	. . . . . . . . . 10 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ ( F ` c ) < U ) -> ( F ` c ) < U )
27	21, 25, 26	elrabd 2888	. . . . . . . . 9 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ ( F ` c ) < U ) -> c e. { w e. ( A [,] B ) | ( F ` w ) < U } )
28	18, 19, 27	rspcdva 2839	. . . . . . . 8 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ ( F ` c ) < U ) -> c < c )
29	17, 28	mtand 660	. . . . . . 7 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> -. ( F ` c ) < U )
30		breq2 3993	. . . . . . . . 9 |- ( r = c -> ( c < r <-> c < c ) )
31		simplrr 531	. . . . . . . . 9 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ U < ( F ` c ) ) -> A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r )
32	20	breq2d 4001	. . . . . . . . . 10 |- ( w = c -> ( U < ( F ` w ) <-> U < ( F ` c ) ) )
33	24	ad2antrr 485	. . . . . . . . . 10 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ U < ( F ` c ) ) -> c e. ( A [,] B ) )
34		simpr 109	. . . . . . . . . 10 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ U < ( F ` c ) ) -> U < ( F ` c ) )
35	32, 33, 34	elrabd 2888	. . . . . . . . 9 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ U < ( F ` c ) ) -> c e. { w e. ( A [,] B ) | U < ( F ` w ) } )
36	30, 31, 35	rspcdva 2839	. . . . . . . 8 |- ( ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) /\ U < ( F ` c ) ) -> c < c )
37	17, 36	mtand 660	. . . . . . 7 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> -. U < ( F ` c ) )
38		ioran 747	. . . . . . 7 |- ( -. ( ( F ` c ) < U \/ U < ( F ` c ) ) <-> ( -. ( F ` c ) < U /\ -. U < ( F ` c ) ) )
39	29, 37, 38	sylanbrc 415	. . . . . 6 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> -. ( ( F ` c ) < U \/ U < ( F ` c ) ) )
40		fveq2 5496	. . . . . . . . . 10 |- ( x = c -> ( F ` x ) = ( F ` c ) )
41	40	eleq1d 2239	. . . . . . . . 9 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
42	7	ralrimiva 2543	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
43	42	adantr 274	. . . . . . . . 9 |- ( ( ph /\ c e. ( A (,) B ) ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
44	41, 43, 24	rspcdva 2839	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
45	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. RR )
46		reaplt 8507	. . . . . . . 8 |- ( ( ( F ` c ) e. RR /\ U e. RR ) -> ( ( F ` c ) # U <-> ( ( F ` c ) < U \/ U < ( F ` c ) ) ) )
47	44, 45, 46	syl2anc 409	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( F ` c ) # U <-> ( ( F ` c ) < U \/ U < ( F ` c ) ) ) )
48	47	adantr 274	. . . . . 6 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> ( ( F ` c ) # U <-> ( ( F ` c ) < U \/ U < ( F ` c ) ) ) )
49	39, 48	mtbird 668	. . . . 5 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> -. ( F ` c ) # U )
50	44	recnd 7948	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. CC )
51	50	adantr 274	. . . . . 6 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> ( F ` c ) e. CC )
52	3	recnd 7948	. . . . . . 7 |- ( ph -> U e. CC )
53	52	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> U e. CC )
54		apti 8541	. . . . . 6 |- ( ( ( F ` c ) e. CC /\ U e. CC ) -> ( ( F ` c ) = U <-> -. ( F ` c ) # U ) )
55	51, 53, 54	syl2anc 409	. . . . 5 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> ( ( F ` c ) = U <-> -. ( F ` c ) # U ) )
56	49, 55	mpbird 166	. . . 4 |- ( ( ( ph /\ c e. ( A (,) B ) ) /\ ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) ) -> ( F ` c ) = U )
57	56	ex 114	. . 3 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) -> ( F ` c ) = U ) )
58	57	reximdva 2572	. 2 |- ( ph -> ( E. c e. ( A (,) B ) ( A. p e. { w e. ( A [,] B ) | ( F ` w ) < U } p < c /\ A. r e. { w e. ( A [,] B ) | U < ( F ` w ) } c < r ) -> E. c e. ( A (,) B ) ( F ` c ) = U ) )
59	14, 58	mpd 13	1 |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   E!wreu 2450   {crab 2452   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   < clt 7954   # cap 8500   (,)cioo 9845   [,]cicc 9848   -cn->ccncf 13351

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-pre-suploc 7895

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-ioo 9849  df-icc 9852  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-cncf 13352

This theorem is referenced by:  ivthdec  13416  reeff1olem  13486