Theorem itg2monolem2 23518

Description: Lemma for itg2mono 23520. (Contributed by Mario Carneiro, 16-Aug-2014.)

Hypotheses

Ref	Expression
itg2mono.1	|- G = ( x e. RR |-> sup ( ran ( n e. NN |-> ( ( F ` n ) ` x ) ) , RR , < ) )
itg2mono.2	|- ( ( ph /\ n e. NN ) -> ( F ` n ) e. MblFn )
itg2mono.3	|- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )
itg2mono.4	|- ( ( ph /\ n e. NN ) -> ( F ` n ) oR <_ ( F ` ( n + 1 ) ) )
itg2mono.5	|- ( ( ph /\ x e. RR ) -> E. y e. RR A. n e. NN ( ( F ` n ) ` x ) <_ y )
itg2mono.6	|- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )
itg2monolem2.7	|- ( ph -> P e. dom S.1 )
itg2monolem2.8	|- ( ph -> P oR <_ G )
itg2monolem2.9	|- ( ph -> -. ( S.1 ` P ) <_ S )

Assertion

Ref	Expression
itg2monolem2	|- ( ph -> S e. RR )

Distinct variable groups:   x, n, y, G   P, n, x, y   n, F, x, y   ph, n, x, y   S, n, x, y

Proof of Theorem itg2monolem2

Step	Hyp	Ref	Expression
1		itg2mono.6	. . 3 |- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )
2		itg2mono.3	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )
3		icossicc 12260	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
4		fss 6056	. . . . . . . 8 |- ( ( ( F ` n ) : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> ( F ` n ) : RR --> ( 0 [,] +oo ) )
5	2, 3, 4	sylancl 694	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,] +oo ) )
6		itg2cl 23499	. . . . . . 7 |- ( ( F ` n ) : RR --> ( 0 [,] +oo ) -> ( S.2 ` ( F ` n ) ) e. RR* )
7	5, 6	syl 17	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( S.2 ` ( F ` n ) ) e. RR* )
8		eqid 2622	. . . . . 6 |- ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) = ( n e. NN |-> ( S.2 ` ( F ` n ) ) )
9	7, 8	fmptd 6385	. . . . 5 |- ( ph -> ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) : NN --> RR* )
10		frn 6053	. . . . 5 |- ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) : NN --> RR* -> ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) C_ RR* )
11	9, 10	syl 17	. . . 4 |- ( ph -> ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) C_ RR* )
12		supxrcl 12145	. . . 4 |- ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) C_ RR* -> sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < ) e. RR* )
13	11, 12	syl 17	. . 3 |- ( ph -> sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < ) e. RR* )
14	1, 13	syl5eqel 2705	. 2 |- ( ph -> S e. RR* )
15		itg2monolem2.7	. . 3 |- ( ph -> P e. dom S.1 )
16		itg1cl 23452	. . 3 |- ( P e. dom S.1 -> ( S.1 ` P ) e. RR )
17	15, 16	syl 17	. 2 |- ( ph -> ( S.1 ` P ) e. RR )
18		mnfxr 10096	. . . 4 |- -oo e. RR*
19	18	a1i 11	. . 3 |- ( ph -> -oo e. RR* )
20		1nn 11031	. . . . 5 |- 1 e. NN
21	5	ralrimiva 2966	. . . . 5 |- ( ph -> A. n e. NN ( F ` n ) : RR --> ( 0 [,] +oo ) )
22		fveq2 6191	. . . . . . 7 |- ( n = 1 -> ( F ` n ) = ( F ` 1 ) )
23	22	feq1d 6030	. . . . . 6 |- ( n = 1 -> ( ( F ` n ) : RR --> ( 0 [,] +oo ) <-> ( F ` 1 ) : RR --> ( 0 [,] +oo ) ) )
24	23	rspcv 3305	. . . . 5 |- ( 1 e. NN -> ( A. n e. NN ( F ` n ) : RR --> ( 0 [,] +oo ) -> ( F ` 1 ) : RR --> ( 0 [,] +oo ) ) )
25	20, 21, 24	mpsyl 68	. . . 4 |- ( ph -> ( F ` 1 ) : RR --> ( 0 [,] +oo ) )
26		itg2cl 23499	. . . 4 |- ( ( F ` 1 ) : RR --> ( 0 [,] +oo ) -> ( S.2 ` ( F ` 1 ) ) e. RR* )
27	25, 26	syl 17	. . 3 |- ( ph -> ( S.2 ` ( F ` 1 ) ) e. RR* )
28		itg2ge0 23502	. . . . 5 |- ( ( F ` 1 ) : RR --> ( 0 [,] +oo ) -> 0 <_ ( S.2 ` ( F ` 1 ) ) )
29	25, 28	syl 17	. . . 4 |- ( ph -> 0 <_ ( S.2 ` ( F ` 1 ) ) )
30		mnflt0 11959	. . . . 5 |- -oo < 0
31		0xr 10086	. . . . . . 7 |- 0 e. RR*
32		xrltletr 11988	. . . . . . 7 |- ( ( -oo e. RR* /\ 0 e. RR* /\ ( S.2 ` ( F ` 1 ) ) e. RR* ) -> ( ( -oo < 0 /\ 0 <_ ( S.2 ` ( F ` 1 ) ) ) -> -oo < ( S.2 ` ( F ` 1 ) ) ) )
33	18, 31, 32	mp3an12 1414	. . . . . 6 |- ( ( S.2 ` ( F ` 1 ) ) e. RR* -> ( ( -oo < 0 /\ 0 <_ ( S.2 ` ( F ` 1 ) ) ) -> -oo < ( S.2 ` ( F ` 1 ) ) ) )
34	27, 33	syl 17	. . . . 5 |- ( ph -> ( ( -oo < 0 /\ 0 <_ ( S.2 ` ( F ` 1 ) ) ) -> -oo < ( S.2 ` ( F ` 1 ) ) ) )
35	30, 34	mpani 712	. . . 4 |- ( ph -> ( 0 <_ ( S.2 ` ( F ` 1 ) ) -> -oo < ( S.2 ` ( F ` 1 ) ) ) )
36	29, 35	mpd 15	. . 3 |- ( ph -> -oo < ( S.2 ` ( F ` 1 ) ) )
37	22	fveq2d 6195	. . . . . . . 8 |- ( n = 1 -> ( S.2 ` ( F ` n ) ) = ( S.2 ` ( F ` 1 ) ) )
38		fvex 6201	. . . . . . . 8 |- ( S.2 ` ( F ` 1 ) ) e. _V
39	37, 8, 38	fvmpt 6282	. . . . . . 7 |- ( 1 e. NN -> ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) ` 1 ) = ( S.2 ` ( F ` 1 ) ) )
40	20, 39	ax-mp 5	. . . . . 6 |- ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) ` 1 ) = ( S.2 ` ( F ` 1 ) )
41		ffn 6045	. . . . . . . 8 |- ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) : NN --> RR* -> ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) Fn NN )
42	9, 41	syl 17	. . . . . . 7 |- ( ph -> ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) Fn NN )
43		fnfvelrn 6356	. . . . . . 7 |- ( ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) Fn NN /\ 1 e. NN ) -> ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) ` 1 ) e. ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) )
44	42, 20, 43	sylancl 694	. . . . . 6 |- ( ph -> ( ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) ` 1 ) e. ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) )
45	40, 44	syl5eqelr 2706	. . . . 5 |- ( ph -> ( S.2 ` ( F ` 1 ) ) e. ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) )
46		supxrub 12154	. . . . 5 |- ( ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) C_ RR* /\ ( S.2 ` ( F ` 1 ) ) e. ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) ) -> ( S.2 ` ( F ` 1 ) ) <_ sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < ) )
47	11, 45, 46	syl2anc 693	. . . 4 |- ( ph -> ( S.2 ` ( F ` 1 ) ) <_ sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < ) )
48	47, 1	syl6breqr 4695	. . 3 |- ( ph -> ( S.2 ` ( F ` 1 ) ) <_ S )
49	19, 27, 14, 36, 48	xrltletrd 11992	. 2 |- ( ph -> -oo < S )
50		itg2monolem2.9	. . . 4 |- ( ph -> -. ( S.1 ` P ) <_ S )
51	17	rexrd 10089	. . . . 5 |- ( ph -> ( S.1 ` P ) e. RR* )
52		xrltnle 10105	. . . . 5 |- ( ( S e. RR* /\ ( S.1 ` P ) e. RR* ) -> ( S < ( S.1 ` P ) <-> -. ( S.1 ` P ) <_ S ) )
53	14, 51, 52	syl2anc 693	. . . 4 |- ( ph -> ( S < ( S.1 ` P ) <-> -. ( S.1 ` P ) <_ S ) )
54	50, 53	mpbird 247	. . 3 |- ( ph -> S < ( S.1 ` P ) )
55		xrltle 11982	. . . 4 |- ( ( S e. RR* /\ ( S.1 ` P ) e. RR* ) -> ( S < ( S.1 ` P ) -> S <_ ( S.1 ` P ) ) )
56	14, 51, 55	syl2anc 693	. . 3 |- ( ph -> ( S < ( S.1 ` P ) -> S <_ ( S.1 ` P ) ) )
57	54, 56	mpd 15	. 2 |- ( ph -> S <_ ( S.1 ` P ) )
58		xrre 12000	. 2 |- ( ( ( S e. RR* /\ ( S.1 ` P ) e. RR ) /\ ( -oo < S /\ S <_ ( S.1 ` P ) ) ) -> S e. RR )
59	14, 17, 49, 57, 58	syl22anc 1327	1 |- ( ph -> S e. RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oRcofr 6896   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   [,)cico 12177   [,]cicc 12178  MblFncmbf 23383   S.1citg1 23384   S.2citg2 23385

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-fal 1489  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390

This theorem is referenced by:  itg2monolem3  23519