Theorem stoweidlem32 40249

Description: If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem32.1	|- F/ t ph
stoweidlem32.2	|- P = ( t e. T |-> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
stoweidlem32.3	|- F = ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
stoweidlem32.4	|- H = ( t e. T |-> Y )
stoweidlem32.5	|- ( ph -> M e. NN )
stoweidlem32.6	|- ( ph -> Y e. RR )
stoweidlem32.7	|- ( ph -> G : ( 1 ... M ) --> A )
stoweidlem32.8	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem32.9	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweidlem32.10	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweidlem32.11	|- ( ( ph /\ f e. A ) -> f : T --> RR )

Assertion

Ref	Expression
stoweidlem32	|- ( ph -> P e. A )

Distinct variable groups:   f, g, i, t, G   A, f, g   f, F, g   T, f, g, i, t   ph, f, g, i   g, H   i, M, t   t, Y, x   x, T   x, A   x, Y   ph, x

Allowed substitution hints:   ph( t)   A( t, i)   P( x, t, f, g, i)   F( x, t, i)   G( x)   H( x, t, f, i)   M( x, f, g)   Y( f, g, i)

Proof of Theorem stoweidlem32

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem32.2	. . 3 |- P = ( t e. T |-> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
2		stoweidlem32.1	. . . 4 |- F/ t ph
3		stoweidlem32.3	. . . . . . . . . . 11 |- F = ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
4		fveq2 6191	. . . . . . . . . . . . 13 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
5	4	sumeq2sdv 14435	. . . . . . . . . . . 12 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
6	5	cbvmptv 4750	. . . . . . . . . . 11 |- ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( s e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
7	3, 6	eqtri 2644	. . . . . . . . . 10 |- F = ( s e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
8	7	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> F = ( s e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
9		fveq2 6191	. . . . . . . . . . 11 |- ( s = t -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` t ) )
10	9	sumeq2sdv 14435	. . . . . . . . . 10 |- ( s = t -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
11	10	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ t e. T ) /\ s = t ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
12		simpr 477	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> t e. T )
13		fzfid 12772	. . . . . . . . . 10 |- ( ( ph /\ t e. T ) -> ( 1 ... M ) e. Fin )
14		simpl 473	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ph )
15		stoweidlem32.7	. . . . . . . . . . . . . 14 |- ( ph -> G : ( 1 ... M ) --> A )
16	15	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. A )
17		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( f = ( G ` i ) -> ( f e. A <-> ( G ` i ) e. A ) )
18	17	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( ( ph /\ f e. A ) <-> ( ph /\ ( G ` i ) e. A ) ) )
19		feq1 6026	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( f : T --> RR <-> ( G ` i ) : T --> RR ) )
20	18, 19	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( f = ( G ` i ) -> ( ( ( ph /\ f e. A ) -> f : T --> RR ) <-> ( ( ph /\ ( G ` i ) e. A ) -> ( G ` i ) : T --> RR ) ) )
21		stoweidlem32.11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ f e. A ) -> f : T --> RR )
22	20, 21	vtoclg 3266	. . . . . . . . . . . . . 14 |- ( ( G ` i ) e. A -> ( ( ph /\ ( G ` i ) e. A ) -> ( G ` i ) : T --> RR ) )
23	16, 22	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( ph /\ ( G ` i ) e. A ) -> ( G ` i ) : T --> RR ) )
24	14, 16, 23	mp2and 715	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
25	24	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
26		simplr 792	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> t e. T )
27	25, 26	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` t ) e. RR )
28	13, 27	fsumrecl 14465	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) e. RR )
29	8, 11, 12, 28	fvmptd 6288	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( F ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
30	29, 28	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
31	30	recnd 10068	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. CC )
32		stoweidlem32.4	. . . . . . . . . . 11 |- H = ( t e. T |-> Y )
33		eqidd 2623	. . . . . . . . . . . 12 |- ( s = t -> Y = Y )
34	33	cbvmptv 4750	. . . . . . . . . . 11 |- ( s e. T |-> Y ) = ( t e. T |-> Y )
35	32, 34	eqtr4i 2647	. . . . . . . . . 10 |- H = ( s e. T |-> Y )
36	35	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> H = ( s e. T |-> Y ) )
37		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ t e. T ) /\ s = t ) -> Y = Y )
38		stoweidlem32.6	. . . . . . . . . 10 |- ( ph -> Y e. RR )
39	38	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> Y e. RR )
40	36, 37, 12, 39	fvmptd 6288	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = Y )
41	40, 39	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. RR )
42	41	recnd 10068	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. CC )
43	31, 42	mulcomd 10061	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) x. ( H ` t ) ) = ( ( H ` t ) x. ( F ` t ) ) )
44	40, 29	oveq12d 6668	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( H ` t ) x. ( F ` t ) ) = ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
45	43, 44	eqtr2d 2657	. . . 4 |- ( ( ph /\ t e. T ) -> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( F ` t ) x. ( H ` t ) ) )
46	2, 45	mpteq2da 4743	. . 3 |- ( ph -> ( t e. T |-> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) ) = ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) )
47	1, 46	syl5eq 2668	. 2 |- ( ph -> P = ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) )
48		stoweidlem32.5	. . . 4 |- ( ph -> M e. NN )
49		stoweidlem32.8	. . . 4 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
50	2, 3, 48, 15, 49, 21	stoweidlem20 40237	. . 3 |- ( ph -> F e. A )
51		stoweidlem32.10	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
52	51	stoweidlem4 40221	. . . . 5 |- ( ( ph /\ Y e. RR ) -> ( t e. T |-> Y ) e. A )
53	38, 52	mpdan 702	. . . 4 |- ( ph -> ( t e. T |-> Y ) e. A )
54	32, 53	syl5eqel 2705	. . 3 |- ( ph -> H e. A )
55		nfmpt1 4747	. . . . . 6 |- F/_ t ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
56	3, 55	nfcxfr 2762	. . . . 5 |- F/_ t F
57	56	nfeq2 2780	. . . 4 |- F/ t f = F
58		nfmpt1 4747	. . . . . 6 |- F/_ t ( t e. T |-> Y )
59	32, 58	nfcxfr 2762	. . . . 5 |- F/_ t H
60	59	nfeq2 2780	. . . 4 |- F/ t g = H
61		stoweidlem32.9	. . . 4 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
62	57, 60, 61	stoweidlem6 40223	. . 3 |- ( ( ph /\ F e. A /\ H e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
63	50, 54, 62	mpd3an23 1426	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
64	47, 63	eqeltrd 2701	1 |- ( ph -> P e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   ...cfz 12326   sum_csu 14416

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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-rp 11833  df-fz 12327  df-fzo 12466  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

This theorem is referenced by:  stoweidlem44  40261