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Theorem ovolicc2lem1 23285
Description: Lemma for ovolicc2 23290. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1 |- ( ph -> A e. RR )
ovolicc.2 |- ( ph -> B e. RR )
ovolicc.3 |- ( ph -> A <_ B )
ovolicc2.4 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
ovolicc2.5 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
ovolicc2.6 |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )
ovolicc2.7 |- ( ph -> ( A [,] B ) C_ U. U )
ovolicc2.8 |- ( ph -> G : U --> NN )
ovolicc2.9 |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )
Assertion
Ref Expression
ovolicc2lem1 |- ( ( ph /\ X e. U ) -> ( P e. X <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
Distinct variable groups:   t, A   t, B   t, F   t, G   ph, t   t, U   t, X
Allowed substitution hints:   P( t)   S( t)
Proof of Theorem ovolicc2lem1
StepHypRef Expression
1 ovolicc2.8 . . . . . 6 |- ( ph -> G : U --> NN )
21ffvelrnda 6359 . . . . 5 |- ( ( ph /\ X e. U ) -> ( G ` X ) e. NN )
3 ovolicc2.5 . . . . . . 7 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
4 inss2 3834 . . . . . . 7 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
5 fss 6056 . . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR ) ) -> F : NN --> ( RR X. RR ) )
63, 4, 5sylancl 694 . . . . . 6 |- ( ph -> F : NN --> ( RR X. RR ) )
7 fvco3 6275 . . . . . 6 |- ( ( F : NN --> ( RR X. RR ) /\ ( G ` X ) e. NN ) -> ( ( (,) o. F ) ` ( G ` X ) ) = ( (,) ` ( F ` ( G ` X ) ) ) )
86, 7sylan 488 . . . . 5 |- ( ( ph /\ ( G ` X ) e. NN ) -> ( ( (,) o. F ) ` ( G ` X ) ) = ( (,) ` ( F ` ( G ` X ) ) ) )
92, 8syldan 487 . . . 4 |- ( ( ph /\ X e. U ) -> ( ( (,) o. F ) ` ( G ` X ) ) = ( (,) ` ( F ` ( G ` X ) ) ) )
10 ovolicc2.9 . . . . . 6 |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )
1110ralrimiva 2966 . . . . 5 |- ( ph -> A. t e. U ( ( (,) o. F ) ` ( G ` t ) ) = t )
12 fveq2 6191 . . . . . . . 8 |- ( t = X -> ( G ` t ) = ( G ` X ) )
1312fveq2d 6195 . . . . . . 7 |- ( t = X -> ( ( (,) o. F ) ` ( G ` t ) ) = ( ( (,) o. F ) ` ( G ` X ) ) )
14 id 22 . . . . . . 7 |- ( t = X -> t = X )
1513, 14eqeq12d 2637 . . . . . 6 |- ( t = X -> ( ( ( (,) o. F ) ` ( G ` t ) ) = t <-> ( ( (,) o. F ) ` ( G ` X ) ) = X ) )
1615rspccva 3308 . . . . 5 |- ( ( A. t e. U ( ( (,) o. F ) ` ( G ` t ) ) = t /\ X e. U ) -> ( ( (,) o. F ) ` ( G ` X ) ) = X )
1711, 16sylan 488 . . . 4 |- ( ( ph /\ X e. U ) -> ( ( (,) o. F ) ` ( G ` X ) ) = X )
186adantr 481 . . . . . . . 8 |- ( ( ph /\ X e. U ) -> F : NN --> ( RR X. RR ) )
1918, 2ffvelrnd 6360 . . . . . . 7 |- ( ( ph /\ X e. U ) -> ( F ` ( G ` X ) ) e. ( RR X. RR ) )
20 1st2nd2 7205 . . . . . . 7 |- ( ( F ` ( G ` X ) ) e. ( RR X. RR ) -> ( F ` ( G ` X ) ) = <. ( 1st ` ( F ` ( G ` X ) ) ) , ( 2nd ` ( F ` ( G ` X ) ) ) >. )
2119, 20syl 17 . . . . . 6 |- ( ( ph /\ X e. U ) -> ( F ` ( G ` X ) ) = <. ( 1st ` ( F ` ( G ` X ) ) ) , ( 2nd ` ( F ` ( G ` X ) ) ) >. )
2221fveq2d 6195 . . . . 5 |- ( ( ph /\ X e. U ) -> ( (,) ` ( F ` ( G ` X ) ) ) = ( (,) ` <. ( 1st ` ( F ` ( G ` X ) ) ) , ( 2nd ` ( F ` ( G ` X ) ) ) >. ) )
23 df-ov 6653 . . . . 5 |- ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) = ( (,) ` <. ( 1st ` ( F ` ( G ` X ) ) ) , ( 2nd ` ( F ` ( G ` X ) ) ) >. )
2422, 23syl6eqr 2674 . . . 4 |- ( ( ph /\ X e. U ) -> ( (,) ` ( F ` ( G ` X ) ) ) = ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) )
259, 17, 243eqtr3d 2664 . . 3 |- ( ( ph /\ X e. U ) -> X = ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) )
2625eleq2d 2687 . 2 |- ( ( ph /\ X e. U ) -> ( P e. X <-> P e. ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
27 xp1st 7198 . . . 4 |- ( ( F ` ( G ` X ) ) e. ( RR X. RR ) -> ( 1st ` ( F ` ( G ` X ) ) ) e. RR )
2819, 27syl 17 . . 3 |- ( ( ph /\ X e. U ) -> ( 1st ` ( F ` ( G ` X ) ) ) e. RR )
29 xp2nd 7199 . . . 4 |- ( ( F ` ( G ` X ) ) e. ( RR X. RR ) -> ( 2nd ` ( F ` ( G ` X ) ) ) e. RR )
3019, 29syl 17 . . 3 |- ( ( ph /\ X e. U ) -> ( 2nd ` ( F ` ( G ` X ) ) ) e. RR )
31 rexr 10085 . . . 4 |- ( ( 1st ` ( F ` ( G ` X ) ) ) e. RR -> ( 1st ` ( F ` ( G ` X ) ) ) e. RR* )
32 rexr 10085 . . . 4 |- ( ( 2nd ` ( F ` ( G ` X ) ) ) e. RR -> ( 2nd ` ( F ` ( G ` X ) ) ) e. RR* )
33 elioo2 12216 . . . 4 |- ( ( ( 1st ` ( F ` ( G ` X ) ) ) e. RR* /\ ( 2nd ` ( F ` ( G ` X ) ) ) e. RR* ) -> ( P e. ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
3431, 32, 33syl2an 494 . . 3 |- ( ( ( 1st ` ( F ` ( G ` X ) ) ) e. RR /\ ( 2nd ` ( F ` ( G ` X ) ) ) e. RR ) -> ( P e. ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
3528, 30, 34syl2anc 693 . 2 |- ( ( ph /\ X e. U ) -> ( P e. ( ( 1st ` ( F ` ( G ` X ) ) ) (,) ( 2nd ` ( F ` ( G ` X ) ) ) ) <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
3626, 35bitrd 268 1 |- ( ( ph /\ X e. U ) -> ( P e. X <-> ( P e. RR /\ ( 1st ` ( F ` ( G ` X ) ) ) < P /\ P < ( 2nd ` ( F ` ( G ` X ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   (,)cioo 12175   [,]cicc 12178   seqcseq 12801   abscabs 13974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179
This theorem is referenced by:  ovolicc2lem2  23286  ovolicc2lem3  23287  ovolicc2lem4  23288
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