Theorem stoweidlem30 29823

Description: This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t_0) = 0, and p > 0 on T - U. Z is used for t₀, P is used for p, ( G ` i ) is used for p_(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem30.1	|- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
stoweidlem30.2	|- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
stoweidlem30.3	|- ( ph -> M e. NN )
stoweidlem30.4	|- ( ph -> G : ( 1 ... M ) --> Q )
stoweidlem30.5	|- ( ( ph /\ f e. A ) -> f : T --> RR )

Assertion

Ref	Expression
stoweidlem30	|- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )

Distinct variable groups:   f, i, T   A, f   f, G   ph, f, i   h, i, t, T   A, h   h, G, t   h, Z   i, M, t   S, i

Allowed substitution hints:   ph( t, h)   A( t, i)   P( t, f, h, i)   Q( t, f, h, i)   S( t, f, h)   G( i)   M( f, h)   Z( t, f, i)

Proof of Theorem stoweidlem30

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2502	. . . . 5 |- ( s = S -> ( s e. T <-> S e. T ) )
2	1	anbi2d 703	. . . 4 |- ( s = S -> ( ( ph /\ s e. T ) <-> ( ph /\ S e. T ) ) )
3		fveq2 5690	. . . . 5 |- ( s = S -> ( P ` s ) = ( P ` S ) )
4		fveq2 5690	. . . . . . 7 |- ( s = S -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` S ) )
5	4	sumeq2sdv 13180	. . . . . 6 |- ( s = S -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) )
6	5	oveq2d 6106	. . . . 5 |- ( s = S -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )
7	3, 6	eqeq12d 2456	. . . 4 |- ( s = S -> ( ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) <-> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
8	2, 7	imbi12d 320	. . 3 |- ( s = S -> ( ( ( ph /\ s e. T ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) ) <-> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) ) )
9		simpr 461	. . . 4 |- ( ( ph /\ s e. T ) -> s e. T )
10		stoweidlem30.3	. . . . . . 7 |- ( ph -> M e. NN )
11	10	nnrecred 10366	. . . . . 6 |- ( ph -> ( 1 / M ) e. RR )
12	11	adantr 465	. . . . 5 |- ( ( ph /\ s e. T ) -> ( 1 / M ) e. RR )
13		fzfid 11794	. . . . . 6 |- ( ( ph /\ s e. T ) -> ( 1 ... M ) e. Fin )
14		stoweidlem30.1	. . . . . . . . 9 |- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
15		stoweidlem30.4	. . . . . . . . 9 |- ( ph -> G : ( 1 ... M ) --> Q )
16		stoweidlem30.5	. . . . . . . . 9 |- ( ( ph /\ f e. A ) -> f : T --> RR )
17	14, 15, 16	stoweidlem15 29808	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( ( G ` i ) ` s ) e. RR /\ 0 <_ ( ( G ` i ) ` s ) /\ ( ( G ` i ) ` s ) <_ 1 ) )
18	17	simp1d 1000	. . . . . . 7 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( G ` i ) ` s ) e. RR )
19	18	an32s 802	. . . . . 6 |- ( ( ( ph /\ s e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` s ) e. RR )
20	13, 19	fsumrecl 13210	. . . . 5 |- ( ( ph /\ s e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) e. RR )
21	12, 20	remulcld 9413	. . . 4 |- ( ( ph /\ s e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR )
22		fveq2 5690	. . . . . . 7 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
23	22	sumeq2sdv 13180	. . . . . 6 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
24	23	oveq2d 6106	. . . . 5 |- ( t = s -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
25		stoweidlem30.2	. . . . 5 |- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
26	24, 25	fvmptg 5771	. . . 4 |- ( ( s e. T /\ ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
27	9, 21, 26	syl2anc 661	. . 3 |- ( ( ph /\ s e. T ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
28	8, 27	vtoclg 3029	. 2 |- ( S e. T -> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
29	28	anabsi7 815	1 |- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2714   {crab 2718   class class class wbr 4291   e. cmpt 4349   -->wf 5413   ` cfv 5417  (class class class)co 6090   RRcr 9280   0cc0 9281   1c1 9282   x. cmul 9286   <_ cle 9418   / cdiv 9992   NNcn 10321   ...cfz 11436   sum_csu 13162

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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359

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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-sum 13163

This theorem is referenced by:  stoweidlem37  29830  stoweidlem38  29831  stoweidlem44  29837