Theorem stoweidlem37 27653

Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [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
stoweidlem37.1	|- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
stoweidlem37.2	|- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
stoweidlem37.3	|- ( ph -> M e. NN )
stoweidlem37.4	|- ( ph -> G : ( 1 ... M ) --> Q )
stoweidlem37.5	|- ( ( ph /\ f e. A ) -> f : T --> RR )
stoweidlem37.6	|- ( ph -> Z e. T )

Assertion

Ref	Expression
stoweidlem37	|- ( ph -> ( P ` Z ) = 0 )

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, t   i, M, t

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

Proof of Theorem stoweidlem37

Step	Hyp	Ref	Expression
1		stoweidlem37.6	. . 3 |- ( ph -> Z e. T )
2		stoweidlem37.1	. . . 4 |- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
3		stoweidlem37.2	. . . 4 |- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
4		stoweidlem37.3	. . . 4 |- ( ph -> M e. NN )
5		stoweidlem37.4	. . . 4 |- ( ph -> G : ( 1 ... M ) --> Q )
6		stoweidlem37.5	. . . 4 |- ( ( ph /\ f e. A ) -> f : T --> RR )
7	2, 3, 4, 5, 6	stoweidlem30 27646	. . 3 |- ( ( ph /\ Z e. T ) -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
8	1, 7	mpdan 650	. 2 |- ( ph -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
9	5	fnvinran 27552	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. Q )
10		fveq1 5686	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( h ` Z ) = ( ( G ` i ) ` Z ) )
11	10	eqeq1d 2412	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( ( h ` Z ) = 0 <-> ( ( G ` i ) ` Z ) = 0 ) )
12		fveq1 5686	. . . . . . . . . . . . 13 |- ( h = ( G ` i ) -> ( h ` t ) = ( ( G ` i ) ` t ) )
13	12	breq2d 4184	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( 0 <_ ( h ` t ) <-> 0 <_ ( ( G ` i ) ` t ) ) )
14	12	breq1d 4182	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( ( h ` t ) <_ 1 <-> ( ( G ` i ) ` t ) <_ 1 ) )
15	13, 14	anbi12d 692	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
16	15	ralbidv 2686	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> A. t e. T ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
17	11, 16	anbi12d 692	. . . . . . . . 9 |- ( h = ( G ` i ) -> ( ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) <-> ( ( ( G ` i ) ` Z ) = 0 /\ A. t e. T ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) ) )
18	17, 2	elrab2 3054	. . . . . . . 8 |- ( ( G ` i ) e. Q <-> ( ( G ` i ) e. A /\ ( ( ( G ` i ) ` Z ) = 0 /\ A. t e. T ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) ) )
19	9, 18	sylib 189	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) e. A /\ ( ( ( G ` i ) ` Z ) = 0 /\ A. t e. T ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) ) )
20	19	simprd 450	. . . . . 6 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( ( G ` i ) ` Z ) = 0 /\ A. t e. T ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
21	20	simpld 446	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` Z ) = 0 )
22	21	sumeq2dv 12452	. . . 4 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = sum_ i e. ( 1 ... M ) 0 )
23		fzfi 11266	. . . . 5 |- ( 1 ... M ) e. Fin
24		olc 374	. . . . 5 |- ( ( 1 ... M ) e. Fin -> ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) )
25		sumz 12471	. . . . 5 |- ( ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) -> sum_ i e. ( 1 ... M ) 0 = 0 )
26	23, 24, 25	mp2b 10	. . . 4 |- sum_ i e. ( 1 ... M ) 0 = 0
27	22, 26	syl6eq 2452	. . 3 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = 0 )
28	27	oveq2d 6056	. 2 |- ( ph -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) = ( ( 1 / M ) x. 0 ) )
29	4	nncnd 9972	. . . 4 |- ( ph -> M e. CC )
30	4	nnne0d 10000	. . . 4 |- ( ph -> M =/= 0 )
31	29, 30	reccld 9739	. . 3 |- ( ph -> ( 1 / M ) e. CC )
32	31	mul01d 9221	. 2 |- ( ph -> ( ( 1 / M ) x. 0 ) = 0 )
33	8, 28, 32	3eqtrd 2440	1 |- ( ph -> ( P ` Z ) = 0 )

Syntax hints:   -> wi 4   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   {crab 2670   C_ wss 3280   class class class wbr 4172   e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   <_ cle 9077   / cdiv 9633   NNcn 9956   ZZ>=cuz 10444   ...cfz 10999   sum_csu 12434

This theorem is referenced by:  stoweidlem44  27660

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435