Theorem itg1addlem2 23464

Description: Lemma for itg1add 23468. The function I represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both i and j are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 23466 and itg1addlem5 23467. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )
itg1add.3	|- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )

Assertion

Ref	Expression
itg1addlem2	|- ( ph -> I : ( RR X. RR ) --> RR )

Distinct variable groups:   i, j, F   i, G, j   ph, i, j

Allowed substitution hints:   I( i, j)

Proof of Theorem itg1addlem2

Step	Hyp	Ref	Expression
1		iffalse 4095	. . . . . . . 8 |- ( -. ( i = 0 /\ j = 0 ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) )
2	1	adantl 482	. . . . . . 7 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) )
3		i1fadd.1	. . . . . . . . . . 11 |- ( ph -> F e. dom S.1 )
4		i1fima 23445	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( `' F " { i } ) e. dom vol )
5	3, 4	syl 17	. . . . . . . . . 10 |- ( ph -> ( `' F " { i } ) e. dom vol )
6		i1fadd.2	. . . . . . . . . . 11 |- ( ph -> G e. dom S.1 )
7		i1fima 23445	. . . . . . . . . . 11 |- ( G e. dom S.1 -> ( `' G " { j } ) e. dom vol )
8	6, 7	syl 17	. . . . . . . . . 10 |- ( ph -> ( `' G " { j } ) e. dom vol )
9		inmbl 23310	. . . . . . . . . 10 |- ( ( ( `' F " { i } ) e. dom vol /\ ( `' G " { j } ) e. dom vol ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) e. dom vol )
10	5, 8, 9	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) e. dom vol )
11	10	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) e. dom vol )
12		mblvol 23298	. . . . . . . 8 |- ( ( ( `' F " { i } ) i^i ( `' G " { j } ) ) e. dom vol -> ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) = ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) )
13	11, 12	syl 17	. . . . . . 7 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) = ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) )
14	2, 13	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) )
15		neorian 2888	. . . . . . 7 |- ( ( i =/= 0 \/ j =/= 0 ) <-> -. ( i = 0 /\ j = 0 ) )
16		inss1 3833	. . . . . . . . . 10 |- ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' F " { i } )
17	16	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' F " { i } ) )
18	5	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( `' F " { i } ) e. dom vol )
19		mblss 23299	. . . . . . . . . 10 |- ( ( `' F " { i } ) e. dom vol -> ( `' F " { i } ) C_ RR )
20	18, 19	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( `' F " { i } ) C_ RR )
21		mblvol 23298	. . . . . . . . . . 11 |- ( ( `' F " { i } ) e. dom vol -> ( vol ` ( `' F " { i } ) ) = ( vol* ` ( `' F " { i } ) ) )
22	18, 21	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( vol ` ( `' F " { i } ) ) = ( vol* ` ( `' F " { i } ) ) )
23	3	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> F e. dom S.1 )
24		simplrl 800	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> i e. RR )
25		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> i =/= 0 )
26		eldifsn 4317	. . . . . . . . . . . 12 |- ( i e. ( RR \ { 0 } ) <-> ( i e. RR /\ i =/= 0 ) )
27	24, 25, 26	sylanbrc 698	. . . . . . . . . . 11 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> i e. ( RR \ { 0 } ) )
28		i1fima2sn 23447	. . . . . . . . . . 11 |- ( ( F e. dom S.1 /\ i e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { i } ) ) e. RR )
29	23, 27, 28	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( vol ` ( `' F " { i } ) ) e. RR )
30	22, 29	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( vol* ` ( `' F " { i } ) ) e. RR )
31		ovolsscl 23254	. . . . . . . . 9 |- ( ( ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' F " { i } ) /\ ( `' F " { i } ) C_ RR /\ ( vol* ` ( `' F " { i } ) ) e. RR ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
32	17, 20, 30, 31	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ i =/= 0 ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
33		inss2 3834	. . . . . . . . . 10 |- ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' G " { j } )
34	33	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' G " { j } ) )
35	6	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. RR /\ j e. RR ) ) -> G e. dom S.1 )
36	35, 7	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( i e. RR /\ j e. RR ) ) -> ( `' G " { j } ) e. dom vol )
37	36	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( `' G " { j } ) e. dom vol )
38		mblss 23299	. . . . . . . . . 10 |- ( ( `' G " { j } ) e. dom vol -> ( `' G " { j } ) C_ RR )
39	37, 38	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( `' G " { j } ) C_ RR )
40		mblvol 23298	. . . . . . . . . . 11 |- ( ( `' G " { j } ) e. dom vol -> ( vol ` ( `' G " { j } ) ) = ( vol* ` ( `' G " { j } ) ) )
41	37, 40	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( vol ` ( `' G " { j } ) ) = ( vol* ` ( `' G " { j } ) ) )
42	6	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> G e. dom S.1 )
43		simplrr 801	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> j e. RR )
44		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> j =/= 0 )
45		eldifsn 4317	. . . . . . . . . . . 12 |- ( j e. ( RR \ { 0 } ) <-> ( j e. RR /\ j =/= 0 ) )
46	43, 44, 45	sylanbrc 698	. . . . . . . . . . 11 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> j e. ( RR \ { 0 } ) )
47		i1fima2sn 23447	. . . . . . . . . . 11 |- ( ( G e. dom S.1 /\ j e. ( RR \ { 0 } ) ) -> ( vol ` ( `' G " { j } ) ) e. RR )
48	42, 46, 47	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( vol ` ( `' G " { j } ) ) e. RR )
49	41, 48	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( vol* ` ( `' G " { j } ) ) e. RR )
50		ovolsscl 23254	. . . . . . . . 9 |- ( ( ( ( `' F " { i } ) i^i ( `' G " { j } ) ) C_ ( `' G " { j } ) /\ ( `' G " { j } ) C_ RR /\ ( vol* ` ( `' G " { j } ) ) e. RR ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
51	34, 39, 49, 50	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ j =/= 0 ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
52	32, 51	jaodan 826	. . . . . . 7 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ ( i =/= 0 \/ j =/= 0 ) ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
53	15, 52	sylan2br 493	. . . . . 6 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> ( vol* ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) e. RR )
54	14, 53	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ ( i e. RR /\ j e. RR ) ) /\ -. ( i = 0 /\ j = 0 ) ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR )
55	54	ex 450	. . . 4 |- ( ( ph /\ ( i e. RR /\ j e. RR ) ) -> ( -. ( i = 0 /\ j = 0 ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR ) )
56		iftrue 4092	. . . . 5 |- ( ( i = 0 /\ j = 0 ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = 0 )
57		0re 10040	. . . . 5 |- 0 e. RR
58	56, 57	syl6eqel 2709	. . . 4 |- ( ( i = 0 /\ j = 0 ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR )
59	55, 58	pm2.61d2 172	. . 3 |- ( ( ph /\ ( i e. RR /\ j e. RR ) ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR )
60	59	ralrimivva 2971	. 2 |- ( ph -> A. i e. RR A. j e. RR if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR )
61		itg1add.3	. . 3 |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )
62	61	fmpt2 7237	. 2 |- ( A. i e. RR A. j e. RR if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) e. RR <-> I : ( RR X. RR ) --> RR )
63	60, 62	sylib 208	1 |- ( ph -> I : ( RR X. RR ) --> RR )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888   |-> cmpt2 6652   RRcr 9935   0cc0 9936   vol*covol 23231   volcvol 23232   S.1citg1 23384

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-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

This theorem is referenced by:  itg1addlem4  23466  itg1addlem5  23467