Theorem mbfmullem2 23491

Description: Lemma for mbfmul 23493. (Contributed by Mario Carneiro, 7-Sep-2014.)

Hypotheses

Ref	Expression
mbfmul.1	|- ( ph -> F e. MblFn )
mbfmul.2	|- ( ph -> G e. MblFn )
mbfmul.3	|- ( ph -> F : A --> RR )
mbfmul.4	|- ( ph -> G : A --> RR )
mbfmul.5	|- ( ph -> P : NN --> dom S.1 )
mbfmul.6	|- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) )
mbfmul.7	|- ( ph -> Q : NN --> dom S.1 )
mbfmul.8	|- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( Q ` n ) ` x ) ) ~~> ( G ` x ) )

Assertion

Ref	Expression
mbfmullem2	|- ( ph -> ( F oF x. G ) e. MblFn )

Distinct variable groups:   x, n, A   P, n, x   ph, n, x   Q, n, x   n, F, x   n, G, x

Proof of Theorem mbfmullem2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmul.3	. . . 4 |- ( ph -> F : A --> RR )
2		ffn 6045	. . . 4 |- ( F : A --> RR -> F Fn A )
3	1, 2	syl 17	. . 3 |- ( ph -> F Fn A )
4		mbfmul.4	. . . 4 |- ( ph -> G : A --> RR )
5		ffn 6045	. . . 4 |- ( G : A --> RR -> G Fn A )
6	4, 5	syl 17	. . 3 |- ( ph -> G Fn A )
7		fdm 6051	. . . . 5 |- ( F : A --> RR -> dom F = A )
8	1, 7	syl 17	. . . 4 |- ( ph -> dom F = A )
9		mbfmul.1	. . . . 5 |- ( ph -> F e. MblFn )
10		mbfdm 23395	. . . . 5 |- ( F e. MblFn -> dom F e. dom vol )
11	9, 10	syl 17	. . . 4 |- ( ph -> dom F e. dom vol )
12	8, 11	eqeltrrd 2702	. . 3 |- ( ph -> A e. dom vol )
13		inidm 3822	. . 3 |- ( A i^i A ) = A
14		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
15		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
16	3, 6, 12, 12, 13, 14, 15	offval 6904	. 2 |- ( ph -> ( F oF x. G ) = ( x e. A |-> ( ( F ` x ) x. ( G ` x ) ) ) )
17		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
18		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
19		1zzd 11408	. . . 4 |- ( ( ph /\ x e. A ) -> 1 e. ZZ )
20		mbfmul.6	. . . 4 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) )
21		nnex 11026	. . . . . 6 |- NN e. _V
22	21	mptex 6486	. . . . 5 |- ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) e. _V
23	22	a1i 11	. . . 4 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) e. _V )
24		mbfmul.8	. . . 4 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( Q ` n ) ` x ) ) ~~> ( G ` x ) )
25		mbfmul.5	. . . . . . . . . . 11 |- ( ph -> P : NN --> dom S.1 )
26	25	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( P ` n ) e. dom S.1 )
27		i1ff 23443	. . . . . . . . . 10 |- ( ( P ` n ) e. dom S.1 -> ( P ` n ) : RR --> RR )
28	26, 27	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( P ` n ) : RR --> RR )
29	28	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( P ` n ) : RR --> RR )
30		mblss 23299	. . . . . . . . . . 11 |- ( A e. dom vol -> A C_ RR )
31	12, 30	syl 17	. . . . . . . . . 10 |- ( ph -> A C_ RR )
32	31	sselda 3603	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. RR )
33	32	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> x e. RR )
34	29, 33	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( ( P ` n ) ` x ) e. RR )
35	34	recnd 10068	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( ( P ` n ) ` x ) e. CC )
36		eqid 2622	. . . . . 6 |- ( n e. NN |-> ( ( P ` n ) ` x ) ) = ( n e. NN |-> ( ( P ` n ) ` x ) )
37	35, 36	fmptd 6385	. . . . 5 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( P ` n ) ` x ) ) : NN --> CC )
38	37	ffvelrnda 6359	. . . 4 |- ( ( ( ph /\ x e. A ) /\ k e. NN ) -> ( ( n e. NN |-> ( ( P ` n ) ` x ) ) ` k ) e. CC )
39		mbfmul.7	. . . . . . . . . . 11 |- ( ph -> Q : NN --> dom S.1 )
40	39	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( Q ` n ) e. dom S.1 )
41		i1ff 23443	. . . . . . . . . 10 |- ( ( Q ` n ) e. dom S.1 -> ( Q ` n ) : RR --> RR )
42	40, 41	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( Q ` n ) : RR --> RR )
43	42	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( Q ` n ) : RR --> RR )
44	43, 33	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( ( Q ` n ) ` x ) e. RR )
45	44	recnd 10068	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ n e. NN ) -> ( ( Q ` n ) ` x ) e. CC )
46		eqid 2622	. . . . . 6 |- ( n e. NN |-> ( ( Q ` n ) ` x ) ) = ( n e. NN |-> ( ( Q ` n ) ` x ) )
47	45, 46	fmptd 6385	. . . . 5 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( Q ` n ) ` x ) ) : NN --> CC )
48	47	ffvelrnda 6359	. . . 4 |- ( ( ( ph /\ x e. A ) /\ k e. NN ) -> ( ( n e. NN |-> ( ( Q ` n ) ` x ) ) ` k ) e. CC )
49		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( P ` n ) = ( P ` k ) )
50	49	fveq1d 6193	. . . . . . . 8 |- ( n = k -> ( ( P ` n ) ` x ) = ( ( P ` k ) ` x ) )
51		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( Q ` n ) = ( Q ` k ) )
52	51	fveq1d 6193	. . . . . . . 8 |- ( n = k -> ( ( Q ` n ) ` x ) = ( ( Q ` k ) ` x ) )
53	50, 52	oveq12d 6668	. . . . . . 7 |- ( n = k -> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) = ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) )
54		eqid 2622	. . . . . . 7 |- ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) = ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) )
55		ovex 6678	. . . . . . 7 |- ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) e. _V
56	53, 54, 55	fvmpt 6282	. . . . . 6 |- ( k e. NN -> ( ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) ` k ) = ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) )
57	56	adantl 482	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ k e. NN ) -> ( ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) ` k ) = ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) )
58		fvex 6201	. . . . . . . 8 |- ( ( P ` k ) ` x ) e. _V
59	50, 36, 58	fvmpt 6282	. . . . . . 7 |- ( k e. NN -> ( ( n e. NN |-> ( ( P ` n ) ` x ) ) ` k ) = ( ( P ` k ) ` x ) )
60		fvex 6201	. . . . . . . 8 |- ( ( Q ` k ) ` x ) e. _V
61	52, 46, 60	fvmpt 6282	. . . . . . 7 |- ( k e. NN -> ( ( n e. NN |-> ( ( Q ` n ) ` x ) ) ` k ) = ( ( Q ` k ) ` x ) )
62	59, 61	oveq12d 6668	. . . . . 6 |- ( k e. NN -> ( ( ( n e. NN |-> ( ( P ` n ) ` x ) ) ` k ) x. ( ( n e. NN |-> ( ( Q ` n ) ` x ) ) ` k ) ) = ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) )
63	62	adantl 482	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ k e. NN ) -> ( ( ( n e. NN |-> ( ( P ` n ) ` x ) ) ` k ) x. ( ( n e. NN |-> ( ( Q ` n ) ` x ) ) ` k ) ) = ( ( ( P ` k ) ` x ) x. ( ( Q ` k ) ` x ) ) )
64	57, 63	eqtr4d 2659	. . . 4 |- ( ( ( ph /\ x e. A ) /\ k e. NN ) -> ( ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) ` k ) = ( ( ( n e. NN |-> ( ( P ` n ) ` x ) ) ` k ) x. ( ( n e. NN |-> ( ( Q ` n ) ` x ) ) ` k ) ) )
65	17, 19, 20, 23, 24, 38, 48, 64	climmul 14363	. . 3 |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) ~~> ( ( F ` x ) x. ( G ` x ) ) )
66	31	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN ) -> A C_ RR )
67	66	resmptd 5452	. . . 4 |- ( ( ph /\ n e. NN ) -> ( ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) |` A ) = ( x e. A |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) )
68		ffn 6045	. . . . . . . 8 |- ( ( P ` n ) : RR --> RR -> ( P ` n ) Fn RR )
69	28, 68	syl 17	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( P ` n ) Fn RR )
70		ffn 6045	. . . . . . . 8 |- ( ( Q ` n ) : RR --> RR -> ( Q ` n ) Fn RR )
71	42, 70	syl 17	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( Q ` n ) Fn RR )
72		reex 10027	. . . . . . . 8 |- RR e. _V
73	72	a1i 11	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> RR e. _V )
74		inidm 3822	. . . . . . 7 |- ( RR i^i RR ) = RR
75		eqidd 2623	. . . . . . 7 |- ( ( ( ph /\ n e. NN ) /\ x e. RR ) -> ( ( P ` n ) ` x ) = ( ( P ` n ) ` x ) )
76		eqidd 2623	. . . . . . 7 |- ( ( ( ph /\ n e. NN ) /\ x e. RR ) -> ( ( Q ` n ) ` x ) = ( ( Q ` n ) ` x ) )
77	69, 71, 73, 73, 74, 75, 76	offval 6904	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( P ` n ) oF x. ( Q ` n ) ) = ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) )
78	26, 40	i1fmul 23463	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( P ` n ) oF x. ( Q ` n ) ) e. dom S.1 )
79		i1fmbf 23442	. . . . . . 7 |- ( ( ( P ` n ) oF x. ( Q ` n ) ) e. dom S.1 -> ( ( P ` n ) oF x. ( Q ` n ) ) e. MblFn )
80	78, 79	syl 17	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( P ` n ) oF x. ( Q ` n ) ) e. MblFn )
81	77, 80	eqeltrrd 2702	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) e. MblFn )
82	12	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN ) -> A e. dom vol )
83		mbfres 23411	. . . . 5 |- ( ( ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) e. MblFn /\ A e. dom vol ) -> ( ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) |` A ) e. MblFn )
84	81, 82, 83	syl2anc 693	. . . 4 |- ( ( ph /\ n e. NN ) -> ( ( x e. RR |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) |` A ) e. MblFn )
85	67, 84	eqeltrrd 2702	. . 3 |- ( ( ph /\ n e. NN ) -> ( x e. A |-> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) ) e. MblFn )
86		ovex 6678	. . . 4 |- ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) e. _V
87	86	a1i 11	. . 3 |- ( ( ph /\ ( n e. NN /\ x e. A ) ) -> ( ( ( P ` n ) ` x ) x. ( ( Q ` n ) ` x ) ) e. _V )
88	17, 18, 65, 85, 87	mbflim 23435	. 2 |- ( ph -> ( x e. A |-> ( ( F ` x ) x. ( G ` x ) ) ) e. MblFn )
89	16, 88	eqeltrd 2701	1 |- ( ph -> ( F oF x. G ) e. MblFn )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   1c1 9937   x. cmul 9941   NNcn 11020   ~~> cli 14215   volcvol 23232  MblFncmbf 23383   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-cc 9257  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-disj 4621  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-omul 7565  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-acn 8768  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-ioc 12180  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-limsup 14202  df-clim 14219  df-rlim 14220  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:  mbfmullem  23492