Theorem itg2mulclem 23513

Description: Lemma for itg2mulc 23514. (Contributed by Mario Carneiro, 8-Jul-2014.)

Hypotheses

Ref	Expression
itg2mulc.2	|- ( ph -> F : RR --> ( 0 [,) +oo ) )
itg2mulc.3	|- ( ph -> ( S.2 ` F ) e. RR )
itg2mulclem.4	|- ( ph -> A e. RR+ )

Assertion

Ref	Expression
itg2mulclem	|- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )

Proof of Theorem itg2mulclem

Dummy variables x f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg2mulc.2	. . . . . . 7 |- ( ph -> F : RR --> ( 0 [,) +oo ) )
2		icossicc 12260	. . . . . . 7 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
3		fss 6056	. . . . . . 7 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> F : RR --> ( 0 [,] +oo ) )
4	1, 2, 3	sylancl 694	. . . . . 6 |- ( ph -> F : RR --> ( 0 [,] +oo ) )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> F : RR --> ( 0 [,] +oo ) )
6		simpr 477	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> f e. dom S.1 )
7		itg2mulclem.4	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rpreccld 11882	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR+ )
9	8	adantr 481	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> ( 1 / A ) e. RR+ )
10	9	rpred 11872	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> ( 1 / A ) e. RR )
11	6, 10	i1fmulc 23470	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( ( RR X. { ( 1 / A ) } ) oF x. f ) e. dom S.1 )
12		itg2ub 23500	. . . . . 6 |- ( ( F : RR --> ( 0 [,] +oo ) /\ ( ( RR X. { ( 1 / A ) } ) oF x. f ) e. dom S.1 /\ ( ( RR X. { ( 1 / A ) } ) oF x. f ) oR <_ F ) -> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) <_ ( S.2 ` F ) )
13	12	3expia 1267	. . . . 5 |- ( ( F : RR --> ( 0 [,] +oo ) /\ ( ( RR X. { ( 1 / A ) } ) oF x. f ) e. dom S.1 ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. f ) oR <_ F -> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) <_ ( S.2 ` F ) ) )
14	5, 11, 13	syl2anc 693	. . . 4 |- ( ( ph /\ f e. dom S.1 ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. f ) oR <_ F -> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) <_ ( S.2 ` F ) ) )
15		i1ff 23443	. . . . . . . . . 10 |- ( f e. dom S.1 -> f : RR --> RR )
16	15	adantl 482	. . . . . . . . 9 |- ( ( ph /\ f e. dom S.1 ) -> f : RR --> RR )
17	16	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( f ` y ) e. RR )
18		rge0ssre 12280	. . . . . . . . . . 11 |- ( 0 [,) +oo ) C_ RR
19		fss 6056	. . . . . . . . . . 11 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ RR ) -> F : RR --> RR )
20	1, 18, 19	sylancl 694	. . . . . . . . . 10 |- ( ph -> F : RR --> RR )
21	20	adantr 481	. . . . . . . . 9 |- ( ( ph /\ f e. dom S.1 ) -> F : RR --> RR )
22	21	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( F ` y ) e. RR )
23	7	rpred 11872	. . . . . . . . 9 |- ( ph -> A e. RR )
24	23	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> A e. RR )
25	7	rpgt0d 11875	. . . . . . . . 9 |- ( ph -> 0 < A )
26	25	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> 0 < A )
27		ledivmul 10899	. . . . . . . 8 |- ( ( ( f ` y ) e. RR /\ ( F ` y ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( ( f ` y ) / A ) <_ ( F ` y ) <-> ( f ` y ) <_ ( A x. ( F ` y ) ) ) )
28	17, 22, 24, 26, 27	syl112anc 1330	. . . . . . 7 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( ( ( f ` y ) / A ) <_ ( F ` y ) <-> ( f ` y ) <_ ( A x. ( F ` y ) ) ) )
29	17	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( f ` y ) e. CC )
30	24	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> A e. CC )
31	7	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ f e. dom S.1 ) -> A e. RR+ )
32	31	rpne0d 11877	. . . . . . . . . 10 |- ( ( ph /\ f e. dom S.1 ) -> A =/= 0 )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> A =/= 0 )
34	29, 30, 33	divrec2d 10805	. . . . . . . 8 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( ( f ` y ) / A ) = ( ( 1 / A ) x. ( f ` y ) ) )
35	34	breq1d 4663	. . . . . . 7 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( ( ( f ` y ) / A ) <_ ( F ` y ) <-> ( ( 1 / A ) x. ( f ` y ) ) <_ ( F ` y ) ) )
36	28, 35	bitr3d 270	. . . . . 6 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( ( f ` y ) <_ ( A x. ( F ` y ) ) <-> ( ( 1 / A ) x. ( f ` y ) ) <_ ( F ` y ) ) )
37	36	ralbidva 2985	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( A. y e. RR ( f ` y ) <_ ( A x. ( F ` y ) ) <-> A. y e. RR ( ( 1 / A ) x. ( f ` y ) ) <_ ( F ` y ) ) )
38		reex 10027	. . . . . . 7 |- RR e. _V
39	38	a1i 11	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> RR e. _V )
40		ovexd 6680	. . . . . 6 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. _V )
41	16	feqmptd 6249	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> f = ( y e. RR |-> ( f ` y ) ) )
42	7	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> A e. RR+ )
43		fconstmpt 5163	. . . . . . . 8 |- ( RR X. { A } ) = ( y e. RR |-> A )
44	43	a1i 11	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> ( RR X. { A } ) = ( y e. RR |-> A ) )
45	1	feqmptd 6249	. . . . . . . 8 |- ( ph -> F = ( y e. RR |-> ( F ` y ) ) )
46	45	adantr 481	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> F = ( y e. RR |-> ( F ` y ) ) )
47	39, 42, 22, 44, 46	offval2 6914	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> ( ( RR X. { A } ) oF x. F ) = ( y e. RR |-> ( A x. ( F ` y ) ) ) )
48	39, 17, 40, 41, 47	ofrfval2 6915	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( f oR <_ ( ( RR X. { A } ) oF x. F ) <-> A. y e. RR ( f ` y ) <_ ( A x. ( F ` y ) ) ) )
49		ovexd 6680	. . . . . 6 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( ( 1 / A ) x. ( f ` y ) ) e. _V )
50	8	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ f e. dom S.1 ) /\ y e. RR ) -> ( 1 / A ) e. RR+ )
51		fconstmpt 5163	. . . . . . . 8 |- ( RR X. { ( 1 / A ) } ) = ( y e. RR |-> ( 1 / A ) )
52	51	a1i 11	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> ( RR X. { ( 1 / A ) } ) = ( y e. RR |-> ( 1 / A ) ) )
53	39, 50, 17, 52, 41	offval2 6914	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> ( ( RR X. { ( 1 / A ) } ) oF x. f ) = ( y e. RR |-> ( ( 1 / A ) x. ( f ` y ) ) ) )
54	39, 49, 22, 53, 46	ofrfval2 6915	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. f ) oR <_ F <-> A. y e. RR ( ( 1 / A ) x. ( f ` y ) ) <_ ( F ` y ) ) )
55	37, 48, 54	3bitr4d 300	. . . 4 |- ( ( ph /\ f e. dom S.1 ) -> ( f oR <_ ( ( RR X. { A } ) oF x. F ) <-> ( ( RR X. { ( 1 / A ) } ) oF x. f ) oR <_ F ) )
56	6, 10	itg1mulc 23471	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) = ( ( 1 / A ) x. ( S.1 ` f ) ) )
57		itg1cl 23452	. . . . . . . . . 10 |- ( f e. dom S.1 -> ( S.1 ` f ) e. RR )
58	57	adantl 482	. . . . . . . . 9 |- ( ( ph /\ f e. dom S.1 ) -> ( S.1 ` f ) e. RR )
59	58	recnd 10068	. . . . . . . 8 |- ( ( ph /\ f e. dom S.1 ) -> ( S.1 ` f ) e. CC )
60	23	adantr 481	. . . . . . . . 9 |- ( ( ph /\ f e. dom S.1 ) -> A e. RR )
61	60	recnd 10068	. . . . . . . 8 |- ( ( ph /\ f e. dom S.1 ) -> A e. CC )
62	59, 61, 32	divrec2d 10805	. . . . . . 7 |- ( ( ph /\ f e. dom S.1 ) -> ( ( S.1 ` f ) / A ) = ( ( 1 / A ) x. ( S.1 ` f ) ) )
63	56, 62	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) = ( ( S.1 ` f ) / A ) )
64	63	breq1d 4663	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) <_ ( S.2 ` F ) <-> ( ( S.1 ` f ) / A ) <_ ( S.2 ` F ) ) )
65		itg2mulc.3	. . . . . . 7 |- ( ph -> ( S.2 ` F ) e. RR )
66	65	adantr 481	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> ( S.2 ` F ) e. RR )
67	25	adantr 481	. . . . . 6 |- ( ( ph /\ f e. dom S.1 ) -> 0 < A )
68		ledivmul 10899	. . . . . 6 |- ( ( ( S.1 ` f ) e. RR /\ ( S.2 ` F ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( ( S.1 ` f ) / A ) <_ ( S.2 ` F ) <-> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) )
69	58, 66, 60, 67, 68	syl112anc 1330	. . . . 5 |- ( ( ph /\ f e. dom S.1 ) -> ( ( ( S.1 ` f ) / A ) <_ ( S.2 ` F ) <-> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) )
70	64, 69	bitr2d 269	. . . 4 |- ( ( ph /\ f e. dom S.1 ) -> ( ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) <-> ( S.1 ` ( ( RR X. { ( 1 / A ) } ) oF x. f ) ) <_ ( S.2 ` F ) ) )
71	14, 55, 70	3imtr4d 283	. . 3 |- ( ( ph /\ f e. dom S.1 ) -> ( f oR <_ ( ( RR X. { A } ) oF x. F ) -> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) )
72	71	ralrimiva 2966	. 2 |- ( ph -> A. f e. dom S.1 ( f oR <_ ( ( RR X. { A } ) oF x. F ) -> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) )
73		ge0mulcl 12285	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
74	73	adantl 482	. . . . 5 |- ( ( ph /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
75		fconstg 6092	. . . . . . 7 |- ( A e. RR+ -> ( RR X. { A } ) : RR --> { A } )
76	7, 75	syl 17	. . . . . 6 |- ( ph -> ( RR X. { A } ) : RR --> { A } )
77		rpre 11839	. . . . . . . . 9 |- ( A e. RR+ -> A e. RR )
78		rpge0 11845	. . . . . . . . 9 |- ( A e. RR+ -> 0 <_ A )
79		elrege0 12278	. . . . . . . . 9 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
80	77, 78, 79	sylanbrc 698	. . . . . . . 8 |- ( A e. RR+ -> A e. ( 0 [,) +oo ) )
81	7, 80	syl 17	. . . . . . 7 |- ( ph -> A e. ( 0 [,) +oo ) )
82	81	snssd 4340	. . . . . 6 |- ( ph -> { A } C_ ( 0 [,) +oo ) )
83	76, 82	fssd 6057	. . . . 5 |- ( ph -> ( RR X. { A } ) : RR --> ( 0 [,) +oo ) )
84	38	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
85		inidm 3822	. . . . 5 |- ( RR i^i RR ) = RR
86	74, 83, 1, 84, 84, 85	off 6912	. . . 4 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
87		fss 6056	. . . 4 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
88	86, 2, 87	sylancl 694	. . 3 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
89	23, 65	remulcld 10070	. . . 4 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR )
90	89	rexrd 10089	. . 3 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR* )
91		itg2leub 23501	. . 3 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) /\ ( A x. ( S.2 ` F ) ) e. RR* ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) <-> A. f e. dom S.1 ( f oR <_ ( ( RR X. { A } ) oF x. F ) -> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) ) )
92	88, 90, 91	syl2anc 693	. 2 |- ( ph -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) <-> A. f e. dom S.1 ( f oR <_ ( ( RR X. { A } ) oF x. F ) -> ( S.1 ` f ) <_ ( A x. ( S.2 ` F ) ) ) ) )
93	72, 92	mpbird 247	1 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   oRcofr 6896   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   RR+crp 11832   [,)cico 12177   [,]cicc 12178   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:  itg2mulc  23514