Theorem ivthinc 14057

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 2177	. . . 4 |- { w e. ( A [,] B ) | ( F ` w ) < U } = { w e. ( A [,] B ) | ( F ` w ) < U }
11		eqid 2177	. . . 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 14056	. . 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 2690	. . 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 9911	. . . . . . . . . 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 8068	. . . . . . . 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 4006	. . . . . . . . 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 5515	. . . . . . . . . . 11 |- ( w = c -> ( F ` w ) = ( F ` c ) )
21	20	breq1d 4013	. . . . . . . . . 10 |- ( w = c -> ( ( F ` w ) < U <-> ( F ` c ) < U ) )
22		ioossicc 9958	. . . . . . . . . . . . 13 |- ( A (,) B ) C_ ( A [,] B )
23	22	sseli 3151	. . . . . . . . . . . 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 2895	. . . . . . . . 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 2846	. . . . . . . 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 665	. . . . . . 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 4007	. . . . . . . . 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 4015	. . . . . . . . . 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 2895	. . . . . . . . 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 2846	. . . . . . . 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 665	. . . . . . 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 752	. . . . . . 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 5515	. . . . . . . . . 10 |- ( x = c -> ( F ` x ) = ( F ` c ) )
41	40	eleq1d 2246	. . . . . . . . 9 |- ( x = c -> ( ( F ` x ) e. RR <-> ( F ` c ) e. RR ) )
42	7	ralrimiva 2550	. . . . . . . . . 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 2846	. . . . . . . 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 8544	. . . . . . . 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 673	. . . . 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 7985	. . . . . . 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 7985	. . . . . . 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 8578	. . . . . 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 2579	. 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 708   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   {crab 2459   C_ wss 3129   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   < clt 7991   # cap 8537   (,)cioo 9887   [,]cicc 9890   -cn->ccncf 13993

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930  ax-pre-suploc 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-rp 9653  df-ioo 9891  df-icc 9894  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-cncf 13994

This theorem is referenced by:  ivthdec  14058  reeff1olem  14128