Theorem stoweidlem32 37893

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 5865	. . . . . . . . . . . . 13 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
5	4	sumeq2sdv 13770	. . . . . . . . . . . 12 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
6	5	cbvmptv 4495	. . . . . . . . . . 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 2473	. . . . . . . . . 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 5865	. . . . . . . . . . 11 |- ( s = t -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` t ) )
10	9	sumeq2sdv 13770	. . . . . . . . . 10 |- ( s = t -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
11	10	adantl 468	. . . . . . . . 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 463	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> t e. T )
13		fzfid 12186	. . . . . . . . . 10 |- ( ( ph /\ t e. T ) -> ( 1 ... M ) e. Fin )
14		simpl 459	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ph )
15		stoweidlem32.7	. . . . . . . . . . . . . 14 |- ( ph -> G : ( 1 ... M ) --> A )
16	15	fnvinran 37335	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. A )
17		eleq1 2517	. . . . . . . . . . . . . . . . 17 |- ( f = ( G ` i ) -> ( f e. A <-> ( G ` i ) e. A ) )
18	17	anbi2d 710	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( ( ph /\ f e. A ) <-> ( ph /\ ( G ` i ) e. A ) ) )
19		feq1 5710	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( f : T --> RR <-> ( G ` i ) : T --> RR ) )
20	18, 19	imbi12d 322	. . . . . . . . . . . . . . 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 3107	. . . . . . . . . . . . . 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 685	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
25	24	adantlr 721	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
26		simplr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> t e. T )
27	25, 26	ffvelrnd 6023	. . . . . . . . . 10 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` t ) e. RR )
28	13, 27	fsumrecl 13800	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) e. RR )
29	8, 11, 12, 28	fvmptd 5954	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( F ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
30	29, 28	eqeltrd 2529	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
31	30	recnd 9669	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. CC )
32		stoweidlem32.4	. . . . . . . . . . 11 |- H = ( t e. T |-> Y )
33		eqidd 2452	. . . . . . . . . . . 12 |- ( s = t -> Y = Y )
34	33	cbvmptv 4495	. . . . . . . . . . 11 |- ( s e. T |-> Y ) = ( t e. T |-> Y )
35	32, 34	eqtr4i 2476	. . . . . . . . . 10 |- H = ( s e. T |-> Y )
36	35	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> H = ( s e. T |-> Y ) )
37		eqidd 2452	. . . . . . . . 9 |- ( ( ( ph /\ t e. T ) /\ s = t ) -> Y = Y )
38		stoweidlem32.6	. . . . . . . . . 10 |- ( ph -> Y e. RR )
39	38	adantr 467	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> Y e. RR )
40	36, 37, 12, 39	fvmptd 5954	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = Y )
41	40, 39	eqeltrd 2529	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. RR )
42	41	recnd 9669	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. CC )
43	31, 42	mulcomd 9664	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) x. ( H ` t ) ) = ( ( H ` t ) x. ( F ` t ) ) )
44	40, 29	oveq12d 6308	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( H ` t ) x. ( F ` t ) ) = ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
45	43, 44	eqtr2d 2486	. . . 4 |- ( ( ph /\ t e. T ) -> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( F ` t ) x. ( H ` t ) ) )
46	2, 45	mpteq2da 4488	. . 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 2497	. 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 37880	. . 3 |- ( ph -> F e. A )
51		stoweidlem32.10	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
52	51	stoweidlem4 37864	. . . . 5 |- ( ( ph /\ Y e. RR ) -> ( t e. T |-> Y ) e. A )
53	38, 52	mpdan 674	. . . 4 |- ( ph -> ( t e. T |-> Y ) e. A )
54	32, 53	syl5eqel 2533	. . 3 |- ( ph -> H e. A )
55		nfmpt1 4492	. . . . . 6 |- F/_ t ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
56	3, 55	nfcxfr 2590	. . . . 5 |- F/_ t F
57	56	nfeq2 2607	. . . 4 |- F/ t f = F
58		nfmpt1 4492	. . . . . 6 |- F/_ t ( t e. T |-> Y )
59	32, 58	nfcxfr 2590	. . . . 5 |- F/_ t H
60	59	nfeq2 2607	. . . 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 37866	. . 3 |- ( ( ph /\ F e. A /\ H e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
63	50, 54, 62	mpd3an23 1366	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
64	47, 63	eqeltrd 2529	1 |- ( ph -> P e. A )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   F/wnf 1667   e. wcel 1887   |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   1c1 9540   + caddc 9542   x. cmul 9544   NNcn 10609   ...cfz 11784   sum_csu 13752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753

This theorem is referenced by:  stoweidlem44  37905