Theorem i1fmulc 23470

Description: A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
i1fmulc.2	|- ( ph -> F e. dom S.1 )
i1fmulc.3	|- ( ph -> A e. RR )

Assertion

Ref	Expression
i1fmulc	|- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )

Proof of Theorem i1fmulc

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

Step	Hyp	Ref	Expression
1		reex 10027	. . . . 5 |- RR e. _V
2	1	a1i 11	. . . 4 |- ( ( ph /\ A = 0 ) -> RR e. _V )
3		i1fmulc.2	. . . . . 6 |- ( ph -> F e. dom S.1 )
4		i1ff 23443	. . . . . 6 |- ( F e. dom S.1 -> F : RR --> RR )
5	3, 4	syl 17	. . . . 5 |- ( ph -> F : RR --> RR )
6	5	adantr 481	. . . 4 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
7		i1fmulc.3	. . . . 5 |- ( ph -> A e. RR )
8	7	adantr 481	. . . 4 |- ( ( ph /\ A = 0 ) -> A e. RR )
9		0red 10041	. . . 4 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
10		simplr 792	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
11	10	oveq1d 6665	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
12		mul02lem2 10213	. . . . . 6 |- ( x e. RR -> ( 0 x. x ) = 0 )
13	12	adantl 482	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
14	11, 13	eqtrd 2656	. . . 4 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
15	2, 6, 8, 9, 14	caofid2 6928	. . 3 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
16		i1f0 23454	. . 3 |- ( RR X. { 0 } ) e. dom S.1
17	15, 16	syl6eqel 2709	. 2 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
18		remulcl 10021	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
19	18	adantl 482	. . . . 5 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
20		fconst6g 6094	. . . . . 6 |- ( A e. RR -> ( RR X. { A } ) : RR --> RR )
21	7, 20	syl 17	. . . . 5 |- ( ph -> ( RR X. { A } ) : RR --> RR )
22	1	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
23		inidm 3822	. . . . 5 |- ( RR i^i RR ) = RR
24	19, 21, 5, 22, 22, 23	off 6912	. . . 4 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
25	24	adantr 481	. . 3 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
26		i1frn 23444	. . . . . . 7 |- ( F e. dom S.1 -> ran F e. Fin )
27	3, 26	syl 17	. . . . . 6 |- ( ph -> ran F e. Fin )
28		ovex 6678	. . . . . . . 8 |- ( A x. y ) e. _V
29		eqid 2622	. . . . . . . 8 |- ( y e. ran F |-> ( A x. y ) ) = ( y e. ran F |-> ( A x. y ) )
30	28, 29	fnmpti 6022	. . . . . . 7 |- ( y e. ran F |-> ( A x. y ) ) Fn ran F
31		dffn4 6121	. . . . . . 7 |- ( ( y e. ran F |-> ( A x. y ) ) Fn ran F <-> ( y e. ran F |-> ( A x. y ) ) : ran F -onto-> ran ( y e. ran F |-> ( A x. y ) ) )
32	30, 31	mpbi 220	. . . . . 6 |- ( y e. ran F |-> ( A x. y ) ) : ran F -onto-> ran ( y e. ran F |-> ( A x. y ) )
33		fofi 8252	. . . . . 6 |- ( ( ran F e. Fin /\ ( y e. ran F |-> ( A x. y ) ) : ran F -onto-> ran ( y e. ran F |-> ( A x. y ) ) ) -> ran ( y e. ran F |-> ( A x. y ) ) e. Fin )
34	27, 32, 33	sylancl 694	. . . . 5 |- ( ph -> ran ( y e. ran F |-> ( A x. y ) ) e. Fin )
35		id 22	. . . . . . . . . . 11 |- ( w e. ran F -> w e. ran F )
36		elsni 4194	. . . . . . . . . . . 12 |- ( x e. { A } -> x = A )
37	36	oveq1d 6665	. . . . . . . . . . 11 |- ( x e. { A } -> ( x x. w ) = ( A x. w ) )
38		oveq2 6658	. . . . . . . . . . . . 13 |- ( y = w -> ( A x. y ) = ( A x. w ) )
39	38	eqeq2d 2632	. . . . . . . . . . . 12 |- ( y = w -> ( ( x x. w ) = ( A x. y ) <-> ( x x. w ) = ( A x. w ) ) )
40	39	rspcev 3309	. . . . . . . . . . 11 |- ( ( w e. ran F /\ ( x x. w ) = ( A x. w ) ) -> E. y e. ran F ( x x. w ) = ( A x. y ) )
41	35, 37, 40	syl2anr 495	. . . . . . . . . 10 |- ( ( x e. { A } /\ w e. ran F ) -> E. y e. ran F ( x x. w ) = ( A x. y ) )
42		ovex 6678	. . . . . . . . . . 11 |- ( x x. w ) e. _V
43		eqeq1 2626	. . . . . . . . . . . 12 |- ( z = ( x x. w ) -> ( z = ( A x. y ) <-> ( x x. w ) = ( A x. y ) ) )
44	43	rexbidv 3052	. . . . . . . . . . 11 |- ( z = ( x x. w ) -> ( E. y e. ran F z = ( A x. y ) <-> E. y e. ran F ( x x. w ) = ( A x. y ) ) )
45	42, 44	elab 3350	. . . . . . . . . 10 |- ( ( x x. w ) e. { z | E. y e. ran F z = ( A x. y ) } <-> E. y e. ran F ( x x. w ) = ( A x. y ) )
46	41, 45	sylibr 224	. . . . . . . . 9 |- ( ( x e. { A } /\ w e. ran F ) -> ( x x. w ) e. { z | E. y e. ran F z = ( A x. y ) } )
47	46	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( x e. { A } /\ w e. ran F ) ) -> ( x x. w ) e. { z | E. y e. ran F z = ( A x. y ) } )
48		fconstg 6092	. . . . . . . . 9 |- ( A e. RR -> ( RR X. { A } ) : RR --> { A } )
49	7, 48	syl 17	. . . . . . . 8 |- ( ph -> ( RR X. { A } ) : RR --> { A } )
50		ffn 6045	. . . . . . . . . 10 |- ( F : RR --> RR -> F Fn RR )
51	5, 50	syl 17	. . . . . . . . 9 |- ( ph -> F Fn RR )
52		dffn3 6054	. . . . . . . . 9 |- ( F Fn RR <-> F : RR --> ran F )
53	51, 52	sylib 208	. . . . . . . 8 |- ( ph -> F : RR --> ran F )
54	47, 49, 53, 22, 22, 23	off 6912	. . . . . . 7 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> { z | E. y e. ran F z = ( A x. y ) } )
55		frn 6053	. . . . . . 7 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> { z | E. y e. ran F z = ( A x. y ) } -> ran ( ( RR X. { A } ) oF x. F ) C_ { z | E. y e. ran F z = ( A x. y ) } )
56	54, 55	syl 17	. . . . . 6 |- ( ph -> ran ( ( RR X. { A } ) oF x. F ) C_ { z | E. y e. ran F z = ( A x. y ) } )
57	29	rnmpt 5371	. . . . . 6 |- ran ( y e. ran F |-> ( A x. y ) ) = { z | E. y e. ran F z = ( A x. y ) }
58	56, 57	syl6sseqr 3652	. . . . 5 |- ( ph -> ran ( ( RR X. { A } ) oF x. F ) C_ ran ( y e. ran F |-> ( A x. y ) ) )
59		ssfi 8180	. . . . 5 |- ( ( ran ( y e. ran F |-> ( A x. y ) ) e. Fin /\ ran ( ( RR X. { A } ) oF x. F ) C_ ran ( y e. ran F |-> ( A x. y ) ) ) -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
60	34, 58, 59	syl2anc 693	. . . 4 |- ( ph -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
61	60	adantr 481	. . 3 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
62		frn 6053	. . . . . . . . 9 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
63	24, 62	syl 17	. . . . . . . 8 |- ( ph -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
64	63	ssdifssd 3748	. . . . . . 7 |- ( ph -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
65	64	adantr 481	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
66	65	sselda 3603	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> y e. RR )
67	3, 7	i1fmulclem 23469	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { y } ) = ( `' F " { ( y / A ) } ) )
68	66, 67	syldan 487	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { y } ) = ( `' F " { ( y / A ) } ) )
69		i1fima 23445	. . . . . 6 |- ( F e. dom S.1 -> ( `' F " { ( y / A ) } ) e. dom vol )
70	3, 69	syl 17	. . . . 5 |- ( ph -> ( `' F " { ( y / A ) } ) e. dom vol )
71	70	ad2antrr 762	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( `' F " { ( y / A ) } ) e. dom vol )
72	68, 71	eqeltrd 2701	. . 3 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { y } ) e. dom vol )
73	68	fveq2d 6195	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { y } ) ) = ( vol ` ( `' F " { ( y / A ) } ) ) )
74	3	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> F e. dom S.1 )
75	7	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. RR )
76		simplr 792	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
77	66, 75, 76	redivcld 10853	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( y / A ) e. RR )
78	66	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> y e. CC )
79	75	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
80		eldifsni 4320	. . . . . . . 8 |- ( y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> y =/= 0 )
81	80	adantl 482	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> y =/= 0 )
82	78, 79, 81, 76	divne0d 10817	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( y / A ) =/= 0 )
83		eldifsn 4317	. . . . . 6 |- ( ( y / A ) e. ( RR \ { 0 } ) <-> ( ( y / A ) e. RR /\ ( y / A ) =/= 0 ) )
84	77, 82, 83	sylanbrc 698	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( y / A ) e. ( RR \ { 0 } ) )
85		i1fima2sn 23447	. . . . 5 |- ( ( F e. dom S.1 /\ ( y / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( y / A ) } ) ) e. RR )
86	74, 84, 85	syl2anc 693	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( y / A ) } ) ) e. RR )
87	73, 86	eqeltrd 2701	. . 3 |- ( ( ( ph /\ A =/= 0 ) /\ y e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { y } ) ) e. RR )
88	25, 61, 72, 87	i1fd 23448	. 2 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
89	17, 88	pm2.61dane 2881	1 |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   RRcr 9935   0cc0 9936   x. cmul 9941   / cdiv 10684   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:  itg1mulc  23471  i1fsub  23475  itg1sub  23476  itg2const  23507  itg2mulclem  23513  itg2monolem1  23517  i1fibl  23574  itgitg1  23575  itg2addnclem  33461  ftc1anclem5  33489