Theorem uniioombllem2a 23350

Description: Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 7-May-2015.)

Hypotheses

Ref	Expression
uniioombl.1	|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
uniioombl.2	|- ( ph -> Disj x e. NN ( (,) ` ( F ` x ) ) )
uniioombl.3	|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
uniioombl.a	|- A = U. ran ( (,) o. F )
uniioombl.e	|- ( ph -> ( vol* ` E ) e. RR )
uniioombl.c	|- ( ph -> C e. RR+ )
uniioombl.g	|- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
uniioombl.s	|- ( ph -> E C_ U. ran ( (,) o. G ) )
uniioombl.t	|- T = seq 1 ( + , ( ( abs o. - ) o. G ) )
uniioombl.v	|- ( ph -> sup ( ran T , RR* , < ) <_ ( ( vol* ` E ) + C ) )

Assertion

Ref	Expression
uniioombllem2a	|- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( (,) ` ( F ` z ) ) i^i ( (,) ` ( G ` J ) ) ) e. ran (,) )

Distinct variable groups:   x, z, F   x, G, z   x, A, z   x, C, z   x, J, z   ph, x, z   x, T, z

Allowed substitution hints:   S( x, z)   E( x, z)

Proof of Theorem uniioombllem2a

Step	Hyp	Ref	Expression
1		inss2 3834	. . . . . . . 8 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
2		uniioombl.1	. . . . . . . . . 10 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
3	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J e. NN ) -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
4	3	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( F ` z ) e. ( <_ i^i ( RR X. RR ) ) )
5	1, 4	sseldi 3601	. . . . . . 7 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( F ` z ) e. ( RR X. RR ) )
6		1st2nd2 7205	. . . . . . 7 |- ( ( F ` z ) e. ( RR X. RR ) -> ( F ` z ) = <. ( 1st ` ( F ` z ) ) , ( 2nd ` ( F ` z ) ) >. )
7	5, 6	syl 17	. . . . . 6 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( F ` z ) = <. ( 1st ` ( F ` z ) ) , ( 2nd ` ( F ` z ) ) >. )
8	7	fveq2d 6195	. . . . 5 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( (,) ` ( F ` z ) ) = ( (,) ` <. ( 1st ` ( F ` z ) ) , ( 2nd ` ( F ` z ) ) >. ) )
9		df-ov 6653	. . . . 5 |- ( ( 1st ` ( F ` z ) ) (,) ( 2nd ` ( F ` z ) ) ) = ( (,) ` <. ( 1st ` ( F ` z ) ) , ( 2nd ` ( F ` z ) ) >. )
10	8, 9	syl6eqr 2674	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( (,) ` ( F ` z ) ) = ( ( 1st ` ( F ` z ) ) (,) ( 2nd ` ( F ` z ) ) ) )
11		uniioombl.g	. . . . . . . . . 10 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
12	11	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ J e. NN ) -> ( G ` J ) e. ( <_ i^i ( RR X. RR ) ) )
13	1, 12	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ J e. NN ) -> ( G ` J ) e. ( RR X. RR ) )
14		1st2nd2 7205	. . . . . . . 8 |- ( ( G ` J ) e. ( RR X. RR ) -> ( G ` J ) = <. ( 1st ` ( G ` J ) ) , ( 2nd ` ( G ` J ) ) >. )
15	13, 14	syl 17	. . . . . . 7 |- ( ( ph /\ J e. NN ) -> ( G ` J ) = <. ( 1st ` ( G ` J ) ) , ( 2nd ` ( G ` J ) ) >. )
16	15	fveq2d 6195	. . . . . 6 |- ( ( ph /\ J e. NN ) -> ( (,) ` ( G ` J ) ) = ( (,) ` <. ( 1st ` ( G ` J ) ) , ( 2nd ` ( G ` J ) ) >. ) )
17		df-ov 6653	. . . . . 6 |- ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) = ( (,) ` <. ( 1st ` ( G ` J ) ) , ( 2nd ` ( G ` J ) ) >. )
18	16, 17	syl6eqr 2674	. . . . 5 |- ( ( ph /\ J e. NN ) -> ( (,) ` ( G ` J ) ) = ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) )
19	18	adantr 481	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( (,) ` ( G ` J ) ) = ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) )
20	10, 19	ineq12d 3815	. . 3 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( (,) ` ( F ` z ) ) i^i ( (,) ` ( G ` J ) ) ) = ( ( ( 1st ` ( F ` z ) ) (,) ( 2nd ` ( F ` z ) ) ) i^i ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) ) )
21		ovolfcl 23235	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ z e. NN ) -> ( ( 1st ` ( F ` z ) ) e. RR /\ ( 2nd ` ( F ` z ) ) e. RR /\ ( 1st ` ( F ` z ) ) <_ ( 2nd ` ( F ` z ) ) ) )
22	3, 21	sylan 488	. . . . . 6 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( 1st ` ( F ` z ) ) e. RR /\ ( 2nd ` ( F ` z ) ) e. RR /\ ( 1st ` ( F ` z ) ) <_ ( 2nd ` ( F ` z ) ) ) )
23	22	simp1d 1073	. . . . 5 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 1st ` ( F ` z ) ) e. RR )
24	23	rexrd 10089	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 1st ` ( F ` z ) ) e. RR* )
25	22	simp2d 1074	. . . . 5 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 2nd ` ( F ` z ) ) e. RR )
26	25	rexrd 10089	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 2nd ` ( F ` z ) ) e. RR* )
27		ovolfcl 23235	. . . . . . . 8 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ J e. NN ) -> ( ( 1st ` ( G ` J ) ) e. RR /\ ( 2nd ` ( G ` J ) ) e. RR /\ ( 1st ` ( G ` J ) ) <_ ( 2nd ` ( G ` J ) ) ) )
28	11, 27	sylan 488	. . . . . . 7 |- ( ( ph /\ J e. NN ) -> ( ( 1st ` ( G ` J ) ) e. RR /\ ( 2nd ` ( G ` J ) ) e. RR /\ ( 1st ` ( G ` J ) ) <_ ( 2nd ` ( G ` J ) ) ) )
29	28	simp1d 1073	. . . . . 6 |- ( ( ph /\ J e. NN ) -> ( 1st ` ( G ` J ) ) e. RR )
30	29	rexrd 10089	. . . . 5 |- ( ( ph /\ J e. NN ) -> ( 1st ` ( G ` J ) ) e. RR* )
31	30	adantr 481	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 1st ` ( G ` J ) ) e. RR* )
32	28	simp2d 1074	. . . . . 6 |- ( ( ph /\ J e. NN ) -> ( 2nd ` ( G ` J ) ) e. RR )
33	32	rexrd 10089	. . . . 5 |- ( ( ph /\ J e. NN ) -> ( 2nd ` ( G ` J ) ) e. RR* )
34	33	adantr 481	. . . 4 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( 2nd ` ( G ` J ) ) e. RR* )
35		iooin 12209	. . . 4 |- ( ( ( ( 1st ` ( F ` z ) ) e. RR* /\ ( 2nd ` ( F ` z ) ) e. RR* ) /\ ( ( 1st ` ( G ` J ) ) e. RR* /\ ( 2nd ` ( G ` J ) ) e. RR* ) ) -> ( ( ( 1st ` ( F ` z ) ) (,) ( 2nd ` ( F ` z ) ) ) i^i ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) ) = ( if ( ( 1st ` ( F ` z ) ) <_ ( 1st ` ( G ` J ) ) , ( 1st ` ( G ` J ) ) , ( 1st ` ( F ` z ) ) ) (,) if ( ( 2nd ` ( F ` z ) ) <_ ( 2nd ` ( G ` J ) ) , ( 2nd ` ( F ` z ) ) , ( 2nd ` ( G ` J ) ) ) ) )
36	24, 26, 31, 34, 35	syl22anc 1327	. . 3 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( ( 1st ` ( F ` z ) ) (,) ( 2nd ` ( F ` z ) ) ) i^i ( ( 1st ` ( G ` J ) ) (,) ( 2nd ` ( G ` J ) ) ) ) = ( if ( ( 1st ` ( F ` z ) ) <_ ( 1st ` ( G ` J ) ) , ( 1st ` ( G ` J ) ) , ( 1st ` ( F ` z ) ) ) (,) if ( ( 2nd ` ( F ` z ) ) <_ ( 2nd ` ( G ` J ) ) , ( 2nd ` ( F ` z ) ) , ( 2nd ` ( G ` J ) ) ) ) )
37	20, 36	eqtrd 2656	. 2 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( (,) ` ( F ` z ) ) i^i ( (,) ` ( G ` J ) ) ) = ( if ( ( 1st ` ( F ` z ) ) <_ ( 1st ` ( G ` J ) ) , ( 1st ` ( G ` J ) ) , ( 1st ` ( F ` z ) ) ) (,) if ( ( 2nd ` ( F ` z ) ) <_ ( 2nd ` ( G ` J ) ) , ( 2nd ` ( F ` z ) ) , ( 2nd ` ( G ` J ) ) ) ) )
38		ioorebas 12275	. 2 |- ( if ( ( 1st ` ( F ` z ) ) <_ ( 1st ` ( G ` J ) ) , ( 1st ` ( G ` J ) ) , ( 1st ` ( F ` z ) ) ) (,) if ( ( 2nd ` ( F ` z ) ) <_ ( 2nd ` ( G ` J ) ) , ( 2nd ` ( F ` z ) ) , ( 2nd ` ( G ` J ) ) ) ) e. ran (,)
39	37, 38	syl6eqel 2709	1 |- ( ( ( ph /\ J e. NN ) /\ z e. NN ) -> ( ( (,) ` ( F ` z ) ) i^i ( (,) ` ( G ` J ) ) ) e. ran (,) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ifcif 4086   <.cop 4183   U.cuni 4436  Disj wdisj 4620   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   supcsup 8346   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   RR+crp 11832   (,)cioo 12175   seqcseq 12801   abscabs 13974   vol*covol 23231

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179

This theorem is referenced by:  uniioombllem2  23351