Theorem stoweidlem32 29832

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 5696	. . . . . . . . . . . . 13 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
5	4	sumeq2sdv 13186	. . . . . . . . . . . 12 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
6	5	cbvmptv 4388	. . . . . . . . . . 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 2463	. . . . . . . . . 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 5696	. . . . . . . . . . 11 |- ( s = t -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` t ) )
10	9	sumeq2sdv 13186	. . . . . . . . . 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 11800	. . . . . . . . . 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 29741	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. A )
17		eleq1 2503	. . . . . . . . . . . . . . . . 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 5547	. . . . . . . . . . . . . . . 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 3035	. . . . . . . . . . . . . 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 5849	. . . . . . . . . 10 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` t ) e. RR )
28	13, 27	fsumrecl 13216	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) e. RR )
29	8, 11, 12, 28	fvmptd 5784	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( F ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
30	29, 28	eqeltrd 2517	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
31	30	recnd 9417	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. CC )
32		stoweidlem32.4	. . . . . . . . . . 11 |- H = ( t e. T |-> Y )
33		eqidd 2444	. . . . . . . . . . . 12 |- ( s = t -> Y = Y )
34	33	cbvmptv 4388	. . . . . . . . . . 11 |- ( s e. T |-> Y ) = ( t e. T |-> Y )
35	32, 34	eqtr4i 2466	. . . . . . . . . 10 |- H = ( s e. T |-> Y )
36	35	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> H = ( s e. T |-> Y ) )
37		eqidd 2444	. . . . . . . . 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 5784	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = Y )
41	40, 39	eqeltrd 2517	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. RR )
42	41	recnd 9417	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. CC )
43	31, 42	mulcomd 9412	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) x. ( H ` t ) ) = ( ( H ` t ) x. ( F ` t ) ) )
44	40, 29	oveq12d 6114	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( H ` t ) x. ( F ` t ) ) = ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
45	43, 44	eqtr2d 2476	. . . 4 |- ( ( ph /\ t e. T ) -> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( F ` t ) x. ( H ` t ) ) )
46	2, 45	mpteq2da 4382	. . 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 2487	. 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 29820	. . 3 |- ( ph -> F e. A )
51		stoweidlem32.10	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
52	51	stoweidlem4 29804	. . . . 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 2527	. . 3 |- ( ph -> H e. A )
55		nfmpt1 4386	. . . . . 6 |- F/_ t ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
56	3, 55	nfcxfr 2581	. . . . 5 |- F/_ t F
57	56	nfeq2 2595	. . . 4 |- F/ t f = F
58		nfmpt1 4386	. . . . . 6 |- F/_ t ( t e. T |-> Y )
59	32, 58	nfcxfr 2581	. . . . 5 |- F/_ t H
60	59	nfeq2 2595	. . . 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 29806	. . 3 |- ( ( ph /\ F e. A /\ H e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
63	50, 54, 62	mpd3an23 1316	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
64	47, 63	eqeltrd 2517	1 |- ( ph -> P e. A )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   F/wnf 1589   e. wcel 1756   e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   1c1 9288   + caddc 9290   x. cmul 9292   NNcn 10327   ...cfz 11442   sum_csu 13168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169

This theorem is referenced by:  stoweidlem44  29844