Theorem ovolicc2lem2 23286

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 )
ovolicc2.10	|- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }
ovolicc2.11	|- ( ph -> H : T --> T )
ovolicc2.12	|- ( ( ph /\ t e. T ) -> if ( ( 2nd ` ( F ` ( G ` t ) ) ) <_ B , ( 2nd ` ( F ` ( G ` t ) ) ) , B ) e. ( H ` t ) )
ovolicc2.13	|- ( ph -> A e. C )
ovolicc2.14	|- ( ph -> C e. T )
ovolicc2.15	|- K = seq 1 ( ( H o. 1st ) , ( NN X. { C } ) )
ovolicc2.16	|- W = { n e. NN | B e. ( K ` n ) }

Assertion

Ref	Expression
ovolicc2lem2	|- ( ( ph /\ ( N e. NN /\ -. N e. W ) ) -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B )

Distinct variable groups:   t, n, u, A   B, n, t, u   t, H   C, n, t   n, F, t   n, K, t, u   n, G, t   n, W   ph, n, t   T, n, t   n, N, t, u   U, n, t, u

Allowed substitution hints:   ph( u)   C( u)   S( u, t, n)   T( u)   F( u)   G( u)   H( u, n)   W( u, t)

Proof of Theorem ovolicc2lem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolicc.2	. . . . . 6 |- ( ph -> B e. RR )
2	1	adantr 481	. . . . 5 |- ( ( ph /\ N e. NN ) -> B e. RR )
3		ovolicc2.5	. . . . . . . . 9 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
4		inss2 3834	. . . . . . . . 9 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
5		fss 6056	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR ) ) -> F : NN --> ( RR X. RR ) )
6	3, 4, 5	sylancl 694	. . . . . . . 8 |- ( ph -> F : NN --> ( RR X. RR ) )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ N e. NN ) -> F : NN --> ( RR X. RR ) )
8		ovolicc2.8	. . . . . . . . 9 |- ( ph -> G : U --> NN )
9	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ N e. NN ) -> G : U --> NN )
10		nnuz 11723	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
11		ovolicc2.15	. . . . . . . . . . . 12 |- K = seq 1 ( ( H o. 1st ) , ( NN X. { C } ) )
12		1zzd 11408	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
13		ovolicc2.14	. . . . . . . . . . . 12 |- ( ph -> C e. T )
14		ovolicc2.11	. . . . . . . . . . . 12 |- ( ph -> H : T --> T )
15	10, 11, 12, 13, 14	algrf 15286	. . . . . . . . . . 11 |- ( ph -> K : NN --> T )
16	15	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ N e. NN ) -> ( K ` N ) e. T )
17		ineq1 3807	. . . . . . . . . . . 12 |- ( u = ( K ` N ) -> ( u i^i ( A [,] B ) ) = ( ( K ` N ) i^i ( A [,] B ) ) )
18	17	neeq1d 2853	. . . . . . . . . . 11 |- ( u = ( K ` N ) -> ( ( u i^i ( A [,] B ) ) =/= (/) <-> ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) ) )
19		ovolicc2.10	. . . . . . . . . . 11 |- T = { u e. U | ( u i^i ( A [,] B ) ) =/= (/) }
20	18, 19	elrab2 3366	. . . . . . . . . 10 |- ( ( K ` N ) e. T <-> ( ( K ` N ) e. U /\ ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) ) )
21	16, 20	sylib 208	. . . . . . . . 9 |- ( ( ph /\ N e. NN ) -> ( ( K ` N ) e. U /\ ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) ) )
22	21	simpld 475	. . . . . . . 8 |- ( ( ph /\ N e. NN ) -> ( K ` N ) e. U )
23	9, 22	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ N e. NN ) -> ( G ` ( K ` N ) ) e. NN )
24	7, 23	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ N e. NN ) -> ( F ` ( G ` ( K ` N ) ) ) e. ( RR X. RR ) )
25		xp2nd 7199	. . . . . 6 |- ( ( F ` ( G ` ( K ` N ) ) ) e. ( RR X. RR ) -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
26	24, 25	syl 17	. . . . 5 |- ( ( ph /\ N e. NN ) -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
27	2, 26	ltnled 10184	. . . 4 |- ( ( ph /\ N e. NN ) -> ( B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <-> -. ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B ) )
28		simprl 794	. . . . . 6 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> N e. NN )
29	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> B e. RR )
30	21	adantrr 753	. . . . . . . . . 10 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( ( K ` N ) e. U /\ ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) ) )
31	30	simprd 479	. . . . . . . . 9 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) )
32		n0 3931	. . . . . . . . 9 |- ( ( ( K ` N ) i^i ( A [,] B ) ) =/= (/) <-> E. x x e. ( ( K ` N ) i^i ( A [,] B ) ) )
33	31, 32	sylib 208	. . . . . . . 8 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> E. x x e. ( ( K ` N ) i^i ( A [,] B ) ) )
34		xp1st 7198	. . . . . . . . . . . 12 |- ( ( F ` ( G ` ( K ` N ) ) ) e. ( RR X. RR ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
35	24, 34	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ N e. NN ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
36	35	adantrr 753	. . . . . . . . . 10 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
37	36	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) e. RR )
38		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> x e. ( ( K ` N ) i^i ( A [,] B ) ) )
39		elin 3796	. . . . . . . . . . . . 13 |- ( x e. ( ( K ` N ) i^i ( A [,] B ) ) <-> ( x e. ( K ` N ) /\ x e. ( A [,] B ) ) )
40	38, 39	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( x e. ( K ` N ) /\ x e. ( A [,] B ) ) )
41	40	simprd 479	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> x e. ( A [,] B ) )
42		ovolicc.1	. . . . . . . . . . . . 13 |- ( ph -> A e. RR )
43		elicc2 12238	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
44	42, 1, 43	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
45	44	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
46	41, 45	mpbid 222	. . . . . . . . . 10 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
47	46	simp1d 1073	. . . . . . . . 9 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> x e. RR )
48	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> B e. RR )
49	40	simpld 475	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> x e. ( K ` N ) )
50	30	simpld 475	. . . . . . . . . . . . 13 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( K ` N ) e. U )
51		ovolicc.3	. . . . . . . . . . . . . 14 |- ( ph -> A <_ B )
52		ovolicc2.4	. . . . . . . . . . . . . 14 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
53		ovolicc2.6	. . . . . . . . . . . . . 14 |- ( ph -> U e. ( ~P ran ( (,) o. F ) i^i Fin ) )
54		ovolicc2.7	. . . . . . . . . . . . . 14 |- ( ph -> ( A [,] B ) C_ U. U )
55		ovolicc2.9	. . . . . . . . . . . . . 14 |- ( ( ph /\ t e. U ) -> ( ( (,) o. F ) ` ( G ` t ) ) = t )
56	42, 1, 51, 52, 3, 53, 54, 8, 55	ovolicc2lem1 23285	. . . . . . . . . . . . 13 |- ( ( ph /\ ( K ` N ) e. U ) -> ( x e. ( K ` N ) <-> ( x e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < x /\ x < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) )
57	50, 56	syldan 487	. . . . . . . . . . . 12 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( x e. ( K ` N ) <-> ( x e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < x /\ x < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) )
58	57	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( x e. ( K ` N ) <-> ( x e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < x /\ x < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) )
59	49, 58	mpbid 222	. . . . . . . . . 10 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( x e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < x /\ x < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) )
60	59	simp2d 1074	. . . . . . . . 9 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < x )
61	46	simp3d 1075	. . . . . . . . 9 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> x <_ B )
62	37, 47, 48, 60, 61	ltletrd 10197	. . . . . . . 8 |- ( ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) /\ x e. ( ( K ` N ) i^i ( A [,] B ) ) ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < B )
63	33, 62	exlimddv 1863	. . . . . . 7 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < B )
64		simprr 796	. . . . . . 7 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) )
65	42, 1, 51, 52, 3, 53, 54, 8, 55	ovolicc2lem1 23285	. . . . . . . 8 |- ( ( ph /\ ( K ` N ) e. U ) -> ( B e. ( K ` N ) <-> ( B e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < B /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) )
66	50, 65	syldan 487	. . . . . . 7 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> ( B e. ( K ` N ) <-> ( B e. RR /\ ( 1st ` ( F ` ( G ` ( K ` N ) ) ) ) < B /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) )
67	29, 63, 64, 66	mpbir3and 1245	. . . . . 6 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> B e. ( K ` N ) )
68		fveq2 6191	. . . . . . . 8 |- ( n = N -> ( K ` n ) = ( K ` N ) )
69	68	eleq2d 2687	. . . . . . 7 |- ( n = N -> ( B e. ( K ` n ) <-> B e. ( K ` N ) ) )
70		ovolicc2.16	. . . . . . 7 |- W = { n e. NN | B e. ( K ` n ) }
71	69, 70	elrab2 3366	. . . . . 6 |- ( N e. W <-> ( N e. NN /\ B e. ( K ` N ) ) )
72	28, 67, 71	sylanbrc 698	. . . . 5 |- ( ( ph /\ ( N e. NN /\ B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) ) ) -> N e. W )
73	72	expr 643	. . . 4 |- ( ( ph /\ N e. NN ) -> ( B < ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) -> N e. W ) )
74	27, 73	sylbird 250	. . 3 |- ( ( ph /\ N e. NN ) -> ( -. ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B -> N e. W ) )
75	74	con1d 139	. 2 |- ( ( ph /\ N e. NN ) -> ( -. N e. W -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B ) )
76	75	impr 649	1 |- ( ( ph /\ ( N e. NN /\ -. N e. W ) ) -> ( 2nd ` ( F ` ( G ` ( K ` N ) ) ) ) <_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   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   < 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-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

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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802

This theorem is referenced by:  ovolicc2lem3  23287  ovolicc2lem4  23288