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Theorem ivthle2 21994
Description: The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
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 )
ivthle2.9  |-  ( ph  ->  ( ( F `  B )  <_  U  /\  U  <_  ( F `
 A ) ) )
Assertion
Ref Expression
ivthle2  |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c
)  =  U )
Distinct variable groups:    x, c, B    D, c, x    F, c, x    ph, c, x    A, c, x    U, c, x

Proof of Theorem ivthle2
StepHypRef Expression
1 ioossicc 11635 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  F  e.  ( D -cn->
CC ) )
14 ivth.8 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  B
)  <  U  /\  U  <  ( F `  A ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth2 21992 . . . . 5  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3561 . . . . 5  |-  ( ( A (,) B ) 
C_  ( A [,] B )  ->  ( E. c  e.  ( A (,) B ) ( F `  c )  =  U  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U ) )
191, 17, 18mpsyl 63 . . . 4  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 648 . . 3  |-  ( ( ( ph  /\  ( F `  B )  <  U )  /\  U  <  ( F `  A
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 9660 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9660 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9750 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 lbicc2 11661 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1228 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
26 eqcom 2466 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5872 . . . . . . . 8  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
2827eqeq2d 2471 . . . . . . 7  |-  ( c  =  A  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  A ) ) )
2926, 28syl5bb 257 . . . . . 6  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  A ) ) )
3029rspcev 3210 . . . . 5  |-  ( ( A  e.  ( A [,] B )  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 471 . . . 4  |-  ( (
ph  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 714 . . 3  |-  ( ( ( ph  /\  ( F `  B )  <  U )  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
33 ivthle2.9 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <_  U  /\  U  <_  ( F `
 A ) ) )
3433simprd 463 . . . . 5  |-  ( ph  ->  U  <_  ( F `  A ) )
3514ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5872 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3736eleq1d 2526 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3837rspcv 3206 . . . . . . 7  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
3925, 35, 38sylc 60 . . . . . 6  |-  ( ph  ->  ( F `  A
)  e.  RR )
406, 39leloed 9745 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  A )  <->  ( U  <  ( F `
 A )  \/  U  =  ( F `
 A ) ) ) )
4134, 40mpbid 210 . . . 4  |-  ( ph  ->  ( U  <  ( F `  A )  \/  U  =  ( F `  A )
) )
4241adantr 465 . . 3  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  ( U  <  ( F `  A
)  \/  U  =  ( F `  A
) ) )
4320, 32, 42mpjaodan 786 . 2  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 ubicc2 11662 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1228 . . 3  |-  ( ph  ->  B  e.  ( A [,] B ) )
46 fveq2 5872 . . . . 5  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
4746eqeq1d 2459 . . . 4  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  ( F `  B )  =  U ) )
4847rspcev 3210 . . 3  |-  ( ( B  e.  ( A [,] B )  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 471 . 2  |-  ( (
ph  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 459 . . 3  |-  ( ph  ->  ( F `  B
)  <_  U )
51 fveq2 5872 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
5251eleq1d 2526 . . . . . 6  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
5352rspcv 3206 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
5445, 35, 53sylc 60 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR )
5554, 6leloed 9745 . . 3  |-  ( ph  ->  ( ( F `  B )  <_  U  <->  ( ( F `  B
)  <  U  \/  ( F `  B )  =  U ) ) )
5650, 55mpbid 210 . 2  |-  ( ph  ->  ( ( F `  B )  <  U  \/  ( F `  B
)  =  U ) )
5743, 49, 56mpjaodan 786 1  |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c
)  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   RR*cxr 9644    < clt 9645    <_ cle 9646   (,)cioo 11554   [,]cicc 11557   -cn->ccncf 21505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cn 19854  df-cnp 19855  df-tx 20188  df-hmeo 20381  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507
This theorem is referenced by:  ivthicc  21995  recosf1o  23047
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