Theorem stoweidlem30 37928

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 2527	. . . . 5 |- ( s = S -> ( s e. T <-> S e. T ) )
2	1	anbi2d 715	. . . 4 |- ( s = S -> ( ( ph /\ s e. T ) <-> ( ph /\ S e. T ) ) )
3		fveq2 5887	. . . . 5 |- ( s = S -> ( P ` s ) = ( P ` S ) )
4		fveq2 5887	. . . . . . 7 |- ( s = S -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` S ) )
5	4	sumeq2sdv 13818	. . . . . 6 |- ( s = S -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) )
6	5	oveq2d 6330	. . . . 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 2476	. . . 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 326	. . 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 467	. . . 4 |- ( ( ph /\ s e. T ) -> s e. T )
10		stoweidlem30.3	. . . . . . 7 |- ( ph -> M e. NN )
11	10	nnrecred 10682	. . . . . 6 |- ( ph -> ( 1 / M ) e. RR )
12	11	adantr 471	. . . . 5 |- ( ( ph /\ s e. T ) -> ( 1 / M ) e. RR )
13		fzfid 12217	. . . . . 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 37912	. . . . . . . 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 1026	. . . . . . 7 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( G ` i ) ` s ) e. RR )
19	18	an32s 818	. . . . . 6 |- ( ( ( ph /\ s e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` s ) e. RR )
20	13, 19	fsumrecl 13848	. . . . 5 |- ( ( ph /\ s e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) e. RR )
21	12, 20	remulcld 9696	. . . 4 |- ( ( ph /\ s e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR )
22		fveq2 5887	. . . . . . 7 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
23	22	sumeq2sdv 13818	. . . . . 6 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
24	23	oveq2d 6330	. . . . 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 5968	. . . 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 671	. . 3 |- ( ( ph /\ s e. T ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
28	8, 27	vtoclg 3118	. 2 |- ( S e. T -> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
29	28	anabsi7 833	1 |- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   {crab 2752   class class class wbr 4415   |-> cmpt 4474   -->wf 5596   ` cfv 5600  (class class class)co 6314   RRcr 9563   0cc0 9564   1c1 9565   x. cmul 9569   <_ cle 9701   / cdiv 10296   NNcn 10636   ...cfz 11812   sum_csu 13800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801

This theorem is referenced by:  stoweidlem37  37935  stoweidlem38  37936  stoweidlem44  37942