Theorem stoweidlem30 31973

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 2529	. . . . 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 5872	. . . . 5 |- ( s = S -> ( P ` s ) = ( P ` S ) )
4		fveq2 5872	. . . . . . 7 |- ( s = S -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` S ) )
5	4	sumeq2sdv 13537	. . . . . 6 |- ( s = S -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) )
6	5	oveq2d 6312	. . . . 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 2479	. . . 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 10602	. . . . . 6 |- ( ph -> ( 1 / M ) e. RR )
12	11	adantr 465	. . . . 5 |- ( ( ph /\ s e. T ) -> ( 1 / M ) e. RR )
13		fzfid 12085	. . . . . 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 31958	. . . . . . . 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 1008	. . . . . . 7 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( G ` i ) ` s ) e. RR )
19	18	an32s 804	. . . . . 6 |- ( ( ( ph /\ s e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` s ) e. RR )
20	13, 19	fsumrecl 13567	. . . . 5 |- ( ( ph /\ s e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) e. RR )
21	12, 20	remulcld 9641	. . . 4 |- ( ( ph /\ s e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR )
22		fveq2 5872	. . . . . . 7 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
23	22	sumeq2sdv 13537	. . . . . 6 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
24	23	oveq2d 6312	. . . . 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 5954	. . . 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 3167	. 2 |- ( S e. T -> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
29	28	anabsi7 819	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 1395   e. wcel 1819   A.wral 2807   {crab 2811   class class class wbr 4456   |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510   x. cmul 9514   <_ cle 9646   / cdiv 10227   NNcn 10556   ...cfz 11697   sum_csu 13519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520

This theorem is referenced by:  stoweidlem37  31980  stoweidlem38  31981  stoweidlem44  31987