Theorem i1fmulclem 23469

Description: Decompose the preimage of a constant times a 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
i1fmulclem	|- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) = ( `' F " { ( B / A ) } ) )

Proof of Theorem i1fmulclem

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

Step	Hyp	Ref	Expression
1		reex 10027	. . . . . . . . . 10 |- RR e. _V
2	1	a1i 11	. . . . . . . . 9 |- ( ph -> RR e. _V )
3		i1fmulc.3	. . . . . . . . 9 |- ( ph -> A e. RR )
4		i1fmulc.2	. . . . . . . . . . 11 |- ( ph -> F e. dom S.1 )
5		i1ff 23443	. . . . . . . . . . 11 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 17	. . . . . . . . . 10 |- ( ph -> F : RR --> RR )
7		ffn 6045	. . . . . . . . . 10 |- ( F : RR --> RR -> F Fn RR )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ph -> F Fn RR )
9		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
10	2, 3, 8, 9	ofc1 6920	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` z ) = ( A x. ( F ` z ) ) )
11	10	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ z e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` z ) = ( A x. ( F ` z ) ) )
12	11	adantlr 751	. . . . . 6 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` z ) = ( A x. ( F ` z ) ) )
13	12	eqeq1d 2624	. . . . 5 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` z ) = B <-> ( A x. ( F ` z ) ) = B ) )
14		eqcom 2629	. . . . . 6 |- ( ( F ` z ) = ( B / A ) <-> ( B / A ) = ( F ` z ) )
15		simplr 792	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> B e. RR )
16	15	recnd 10068	. . . . . . 7 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> B e. CC )
17	3	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> A e. RR )
18	17	recnd 10068	. . . . . . 7 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> A e. CC )
19	6	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> F : RR --> RR )
20	19	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( F ` z ) e. RR )
21	20	recnd 10068	. . . . . . 7 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( F ` z ) e. CC )
22		simpllr 799	. . . . . . 7 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> A =/= 0 )
23	16, 18, 21, 22	divmuld 10823	. . . . . 6 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( ( B / A ) = ( F ` z ) <-> ( A x. ( F ` z ) ) = B ) )
24	14, 23	syl5bb 272	. . . . 5 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( ( F ` z ) = ( B / A ) <-> ( A x. ( F ` z ) ) = B ) )
25	13, 24	bitr4d 271	. . . 4 |- ( ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) /\ z e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` z ) = B <-> ( F ` z ) = ( B / A ) ) )
26	25	pm5.32da 673	. . 3 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( ( z e. RR /\ ( ( ( RR X. { A } ) oF x. F ) ` z ) = B ) <-> ( z e. RR /\ ( F ` z ) = ( B / A ) ) ) )
27		remulcl 10021	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
28	27	adantl 482	. . . . . . 7 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
29		fconstg 6092	. . . . . . . . 9 |- ( A e. RR -> ( RR X. { A } ) : RR --> { A } )
30	3, 29	syl 17	. . . . . . . 8 |- ( ph -> ( RR X. { A } ) : RR --> { A } )
31	3	snssd 4340	. . . . . . . 8 |- ( ph -> { A } C_ RR )
32	30, 31	fssd 6057	. . . . . . 7 |- ( ph -> ( RR X. { A } ) : RR --> RR )
33		inidm 3822	. . . . . . 7 |- ( RR i^i RR ) = RR
34	28, 32, 6, 2, 2, 33	off 6912	. . . . . 6 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
35	34	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
36		ffn 6045	. . . . 5 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ( ( RR X. { A } ) oF x. F ) Fn RR )
37	35, 36	syl 17	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( ( RR X. { A } ) oF x. F ) Fn RR )
38		fniniseg 6338	. . . 4 |- ( ( ( RR X. { A } ) oF x. F ) Fn RR -> ( z e. ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) <-> ( z e. RR /\ ( ( ( RR X. { A } ) oF x. F ) ` z ) = B ) ) )
39	37, 38	syl 17	. . 3 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( z e. ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) <-> ( z e. RR /\ ( ( ( RR X. { A } ) oF x. F ) ` z ) = B ) ) )
40	19, 7	syl 17	. . . 4 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> F Fn RR )
41		fniniseg 6338	. . . 4 |- ( F Fn RR -> ( z e. ( `' F " { ( B / A ) } ) <-> ( z e. RR /\ ( F ` z ) = ( B / A ) ) ) )
42	40, 41	syl 17	. . 3 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( z e. ( `' F " { ( B / A ) } ) <-> ( z e. RR /\ ( F ` z ) = ( B / A ) ) ) )
43	26, 39, 42	3bitr4d 300	. 2 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( z e. ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) <-> z e. ( `' F " { ( B / A ) } ) ) )
44	43	eqrdv 2620	1 |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) = ( `' F " { ( B / A ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {csn 4177   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   RRcr 9935   0cc0 9936   x. cmul 9941   / cdiv 10684   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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-sum 14417  df-itg1 23389

This theorem is referenced by:  i1fmulc  23470  itg1mulc  23471