Theorem ivthinc 15508

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 2232	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } = { w e. ( A [,] B ) | ( F ` w ) < U }
11		eqid 2232	. . . 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 15507	. . 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 2763	. . 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 10245	. . . . . . . . . 10 |- ( c e. ( A (,) B ) -> c e. RR )
16	15	ad2antlr 489	. . . . . . . . 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 8385	. . . . . . . 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 4112	. . . . . . . . 9 |- ( p = c -> ( p < c <-> c < c ) )
19		simplrl 537	. . . . . . . . 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 5670	. . . . . . . . . . 11 |- ( w = c -> ( F ` w ) = ( F ` c ) )
21	20	breq1d 4119	. . . . . . . . . 10 |- ( w = c -> ( ( F ` w ) < U <-> ( F ` c ) < U ) )
22		ioossicc 10292	. . . . . . . . . . . . 13 |- ( A (,) B ) C_ ( A [,] B )
23	22	sseli 3234	. . . . . . . . . . . 12 |- ( c e. ( A (,) B ) -> c e. ( A [,] B ) )
24	23	adantl 277	. . . . . . . . . . 11 |- ( ( ph /\ c e. ( A (,) B ) ) -> c e. ( A [,] B ) )
25	24	ad2antrr 488	. . . . . . . . . 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 110	. . . . . . . . . 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 2975	. . . . . . . . 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 2926	. . . . . . . 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 671	. . . . . . 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 4113	. . . . . . . . 9 |- ( r = c -> ( c < r <-> c < c ) )
31		simplrr 538	. . . . . . . . 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 4121	. . . . . . . . . 10 |- ( w = c -> ( U < ( F ` w ) <-> U < ( F ` c ) ) )
33	24	ad2antrr 488	. . . . . . . . . 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 110	. . . . . . . . . 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 2975	. . . . . . . . 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 2926	. . . . . . . 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 671	. . . . . . 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 760	. . . . . . 7 |- ( -. ( ( F ` c ) < U \/ U < ( F ` c ) ) <-> ( -. ( F ` c ) < U /\ -. U < ( F ` c ) ) )
39	29, 37, 38	sylanbrc 417	. . . . . 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 5670	. . . . . . . . . 10 |- ( x = c -> ( F ` x ) = ( F ` c ) )
41	40	eleq1d 2301	. . . . . . . . 9 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
42	7	ralrimiva 2615	. . . . . . . . . 10 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
43	42	adantr 276	. . . . . . . . 9 |- ( ( ph /\ c e. ( A (,) B ) ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
44	41, 43, 24	rspcdva 2926	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. RR )
45	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ c e. ( A (,) B ) ) -> U e. RR )
46		reaplt 8862	. . . . . . . 8 |- ( ( ( F ` c ) e. RR /\ U e. RR ) -> ( ( F ` c ) # U <-> ( ( F ` c ) < U \/ U < ( F ` c ) ) ) )
47	44, 45, 46	syl2anc 411	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( ( F ` c ) # U <-> ( ( F ` c ) < U \/ U < ( F ` c ) ) ) )
48	47	adantr 276	. . . . . 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 680	. . . . 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 8302	. . . . . . 7 |- ( ( ph /\ c e. ( A (,) B ) ) -> ( F ` c ) e. CC )
51	50	adantr 276	. . . . . 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 8302	. . . . . . 7 |- ( ph -> U e. CC )
53	52	ad2antrr 488	. . . . . 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 8896	. . . . . 6 |- ( ( ( F ` c ) e. CC /\ U e. CC ) -> ( ( F ` c ) = U <-> -. ( F ` c ) # U ) )
55	51, 53, 54	syl2anc 411	. . . . 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 167	. . . 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 115	. . 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 2644	. 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 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   E!wreu 2522   {crab 2524   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   < clt 8308   # cap 8855   (,)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:  ivthdec  15509  reeff1olem  15636