Theorem ivthinc 14797

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 2193	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } = { w e. ( A [,] B ) | ( F ` w ) < U }
11		eqid 2193	. . . 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 14796	. . 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 2712	. . 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 9978	. . . . . . . . . 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 8131	. . . . . . . 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 4032	. . . . . . . . 9 |- ( p = c -> ( p < c <-> c < c ) )
19		simplrl 535	. . . . . . . . 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 5554	. . . . . . . . . . 11 |- ( w = c -> ( F ` w ) = ( F ` c ) )
21	20	breq1d 4039	. . . . . . . . . 10 |- ( w = c -> ( ( F ` w ) < U <-> ( F ` c ) < U ) )
22		ioossicc 10025	. . . . . . . . . . . . 13 |- ( A (,) B ) C_ ( A [,] B )
23	22	sseli 3175	. . . . . . . . . . . 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 2918	. . . . . . . . 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 2869	. . . . . . . 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 666	. . . . . . 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 4033	. . . . . . . . 9 |- ( r = c -> ( c < r <-> c < c ) )
31		simplrr 536	. . . . . . . . 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 4041	. . . . . . . . . 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 2918	. . . . . . . . 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 2869	. . . . . . . 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 666	. . . . . . 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 753	. . . . . . 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 5554	. . . . . . . . . 10 |- ( x = c -> ( F ` x ) = ( F ` c ) )
41	40	eleq1d 2262	. . . . . . . . 9 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
42	7	ralrimiva 2567	. . . . . . . . . 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 2869	. . . . . . . 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 8607	. . . . . . . 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 674	. . . . 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 8048	. . . . . . 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 8048	. . . . . . 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 8641	. . . . . 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 2596	. 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 709   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   E!wreu 2474   {crab 2476   C_ wss 3153   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   < clt 8054   # cap 8600   (,)cioo 9954   [,]cicc 9957   -cn->ccncf 14725

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-pre-suploc 7993

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-ioo 9958  df-icc 9961  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-cncf 14726

This theorem is referenced by:  ivthdec  14798  reeff1olem  14906