Theorem stoweidlem32 38005

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 5879	. . . . . . . . . . . . 13 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
5	4	sumeq2sdv 13847	. . . . . . . . . . . 12 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
6	5	cbvmptv 4488	. . . . . . . . . . 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 2493	. . . . . . . . . 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 5879	. . . . . . . . . . 11 |- ( s = t -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` t ) )
10	9	sumeq2sdv 13847	. . . . . . . . . 10 |- ( s = t -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
11	10	adantl 473	. . . . . . . . 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 468	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> t e. T )
13		fzfid 12224	. . . . . . . . . 10 |- ( ( ph /\ t e. T ) -> ( 1 ... M ) e. Fin )
14		simpl 464	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ph )
15		stoweidlem32.7	. . . . . . . . . . . . . 14 |- ( ph -> G : ( 1 ... M ) --> A )
16	15	fnvinran 37398	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. A )
17		eleq1 2537	. . . . . . . . . . . . . . . . 17 |- ( f = ( G ` i ) -> ( f e. A <-> ( G ` i ) e. A ) )
18	17	anbi2d 718	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( ( ph /\ f e. A ) <-> ( ph /\ ( G ` i ) e. A ) ) )
19		feq1 5720	. . . . . . . . . . . . . . . 16 |- ( f = ( G ` i ) -> ( f : T --> RR <-> ( G ` i ) : T --> RR ) )
20	18, 19	imbi12d 327	. . . . . . . . . . . . . . 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 3093	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
25	24	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( G ` i ) : T --> RR )
26		simplr 770	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> t e. T )
27	25, 26	ffvelrnd 6038	. . . . . . . . . 10 |- ( ( ( ph /\ t e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` t ) e. RR )
28	13, 27	fsumrecl 13877	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) e. RR )
29	8, 11, 12, 28	fvmptd 5969	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( F ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
30	29, 28	eqeltrd 2549	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
31	30	recnd 9687	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. CC )
32		stoweidlem32.4	. . . . . . . . . . 11 |- H = ( t e. T |-> Y )
33		eqidd 2472	. . . . . . . . . . . 12 |- ( s = t -> Y = Y )
34	33	cbvmptv 4488	. . . . . . . . . . 11 |- ( s e. T |-> Y ) = ( t e. T |-> Y )
35	32, 34	eqtr4i 2496	. . . . . . . . . 10 |- H = ( s e. T |-> Y )
36	35	a1i 11	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> H = ( s e. T |-> Y ) )
37		eqidd 2472	. . . . . . . . 9 |- ( ( ( ph /\ t e. T ) /\ s = t ) -> Y = Y )
38		stoweidlem32.6	. . . . . . . . . 10 |- ( ph -> Y e. RR )
39	38	adantr 472	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> Y e. RR )
40	36, 37, 12, 39	fvmptd 5969	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = Y )
41	40, 39	eqeltrd 2549	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. RR )
42	41	recnd 9687	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( H ` t ) e. CC )
43	31, 42	mulcomd 9682	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( F ` t ) x. ( H ` t ) ) = ( ( H ` t ) x. ( F ` t ) ) )
44	40, 29	oveq12d 6326	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( H ` t ) x. ( F ` t ) ) = ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
45	43, 44	eqtr2d 2506	. . . 4 |- ( ( ph /\ t e. T ) -> ( Y x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( F ` t ) x. ( H ` t ) ) )
46	2, 45	mpteq2da 4481	. . 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 2517	. 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 37992	. . 3 |- ( ph -> F e. A )
51		stoweidlem32.10	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
52	51	stoweidlem4 37976	. . . . 5 |- ( ( ph /\ Y e. RR ) -> ( t e. T |-> Y ) e. A )
53	38, 52	mpdan 681	. . . 4 |- ( ph -> ( t e. T |-> Y ) e. A )
54	32, 53	syl5eqel 2553	. . 3 |- ( ph -> H e. A )
55		nfmpt1 4485	. . . . . 6 |- F/_ t ( t e. T |-> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) )
56	3, 55	nfcxfr 2610	. . . . 5 |- F/_ t F
57	56	nfeq2 2627	. . . 4 |- F/ t f = F
58		nfmpt1 4485	. . . . . 6 |- F/_ t ( t e. T |-> Y )
59	32, 58	nfcxfr 2610	. . . . 5 |- F/_ t H
60	59	nfeq2 2627	. . . 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 37978	. . 3 |- ( ( ph /\ F e. A /\ H e. A ) -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
63	50, 54, 62	mpd3an23 1392	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) x. ( H ` t ) ) ) e. A )
64	47, 63	eqeltrd 2549	1 |- ( ph -> P e. A )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   F/wnf 1675   e. wcel 1904   |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   1c1 9558   + caddc 9560   x. cmul 9562   NNcn 10631   ...cfz 11810   sum_csu 13829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830

This theorem is referenced by:  stoweidlem44  38017