Theorem stoweidlem32 31332

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 5864	. . . . . . . . . . . . 13 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
5	4	sumeq2sdv 13482	. . . . . . . . . . . 12 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
6	5	cbvmptv 4538	. . . . . . . . . . 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 2496	. . . . . . . . . 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 5864	. . . . . . . . . . 11 |- ( s = t -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` t ) )
10	9	sumeq2sdv 13482	. . . . . . . . . 10 |- ( s = t -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
11	10	adantl 466	. . . . . . . . 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 461	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> t e. T )
13		fzfid 12046	. . . . . . . . . 10 |- ( ( ph /\ t e. T ) -> ( 1 ... M ) e. Fin )
14		simpl 457	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ph )
15		stoweidlem32.7	. . . . . . . . . . . . . 14 |- ( ph -> G : ( 1 ... M ) --> A )
16	15	fnvinran 30967	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. A )
17		eleq1 2539	. . . . . . . . . . . . . . . . 17 |- ( f = ( G ` i ) -> ( f e. A <-> ( G ` i ) e. A ) )
18	17	anbi2d 703	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( ( ph /\ f e. A ) <-> ( ph /\ ( G ` i ) e. A ) ) )
19		feq1 5711	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( f : T --> RR <-> ( G ` i ) : T --> RR ) )
20	18, 19	imbi12d 320	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . 14 |- ( ( G ` i ) e. A -> ( ( ph /\ ( G ` i ) e. A ) -> ( G ` i ) : T --> RR ) )
23	16, 22	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( ph /\ ( G ` i ) e. A ) -> ( G ` i ) : T --> RR ) )
24	14, 16, 23	mp2and 679	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
25	24	adantlr 714	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
26		simplr 754	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> t e. T )
27	25, 26	ffvelrnd 6020	. . . . . . . . . 10 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` t ) e. RR )
28	13, 27	fsumrecl 13512	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) e. RR )
29	8, 11, 12, 28	fvmptd 5953	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( F ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
30	29, 28	eqeltrd 2555	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
31	30	recnd 9618	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. CC )
32		stoweidlem32.4	. . . . . . . . . . 11 |- H = ( t e. T |-> Y )
33		eqidd 2468	. . . . . . . . . . . 12 |- ( s = t -> Y = Y )
34	33	cbvmptv 4538	. . . . . . . . . . 11 |- ( s e. T |-> Y ) = ( t e. T |-> Y )
35	32, 34	eqtr4i 2499	. . . . . . . . . 10 |- H = ( s e. T |-> Y )
36	35	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> H = ( s e. T |-> Y ) )
37		eqidd 2468	. . . . . . . . 9 |- ( ( ( ph /\ t e. T ) /\ s = t ) -> Y = Y )
38		stoweidlem32.6	. . . . . . . . . 10 |- ( ph -> Y e. RR )
39	38	adantr 465	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> Y e. RR )
40	36, 37, 12, 39	fvmptd 5953	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = Y )
41	40, 39	eqeltrd 2555	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. RR )
42	41	recnd 9618	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. CC )
43	31, 42	mulcomd 9613	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) x. ( H ` t ) ) = ( ( H ` t ) x. ( F ` t ) ) )
44	40, 29	oveq12d 6300	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( H ` t ) x. ( F ` t ) ) = ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
45	43, 44	eqtr2d 2509	. . . 4 |- ( ( ph /\ t e. T ) -> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( F ` t ) x. ( H ` t ) ) )
46	2, 45	mpteq2da 4532	. . 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 2520	. 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 31320	. . 3 |- ( ph -> F e. A )
51		stoweidlem32.10	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
52	51	stoweidlem4 31304	. . . . 5 |- ( ( ph /\ Y e. RR ) -> ( t e. T |-> Y ) e. A )
53	38, 52	mpdan 668	. . . 4 |- ( ph -> ( t e. T |-> Y ) e. A )
54	32, 53	syl5eqel 2559	. . 3 |- ( ph -> H e. A )
55		nfmpt1 4536	. . . . . 6 |- F/_ t ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
56	3, 55	nfcxfr 2627	. . . . 5 |- F/_ t F
57	56	nfeq2 2646	. . . 4 |- F/ t f = F
58		nfmpt1 4536	. . . . . 6 |- F/_ t ( t e. T |-> Y )
59	32, 58	nfcxfr 2627	. . . . 5 |- F/_ t H
60	59	nfeq2 2646	. . . 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 31306	. . 3 |- ( ( ph /\ F e. A /\ H e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
63	50, 54, 62	mpd3an23 1326	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
64	47, 63	eqeltrd 2555	1 |- ( ph -> P e. A )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   F/wnf 1599   e. wcel 1767   |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   1c1 9489   + caddc 9491   x. cmul 9493   NNcn 10532   ...cfz 11668   sum_csu 13464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465

This theorem is referenced by:  stoweidlem44  31344