Theorem stoweidlem37 31337

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 31330	. . 3 |- ( ( ph /\ Z e. T ) -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
8	1, 7	mpdan 668	. 2 |- ( ph -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
9	5	fnvinran 30967	. . . . . . . 8 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. Q )
10		fveq1 5863	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( h ` Z ) = ( ( G ` i ) ` Z ) )
11	10	eqeq1d 2469	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( ( h ` Z ) = 0 <-> ( ( G ` i ) ` Z ) = 0 ) )
12		fveq1 5863	. . . . . . . . . . . . 13 |- ( h = ( G ` i ) -> ( h ` t ) = ( ( G ` i ) ` t ) )
13	12	breq2d 4459	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( 0 <_ ( h ` t ) <-> 0 <_ ( ( G ` i ) ` t ) ) )
14	12	breq1d 4457	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( ( h ` t ) <_ 1 <-> ( ( G ` i ) ` t ) <_ 1 ) )
15	13, 14	anbi12d 710	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
16	15	ralbidv 2903	. . . . . . . . . 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 710	. . . . . . . . 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 3263	. . . . . . . 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 196	. . . . . . 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 463	. . . . . 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 459	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` Z ) = 0 )
22	21	sumeq2dv 13484	. . . 4 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = sum_ i e. ( 1 ... M ) 0 )
23		fzfi 12046	. . . . 5 |- ( 1 ... M ) e. Fin
24		olc 384	. . . . 5 |- ( ( 1 ... M ) e. Fin -> ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) )
25		sumz 13503	. . . . 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 2524	. . 3 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = 0 )
28	27	oveq2d 6298	. 2 |- ( ph -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) = ( ( 1 / M ) x. 0 ) )
29	4	nncnd 10548	. . . 4 |- ( ph -> M e. CC )
30	4	nnne0d 10576	. . . 4 |- ( ph -> M =/= 0 )
31	29, 30	reccld 10309	. . 3 |- ( ph -> ( 1 / M ) e. CC )
32	31	mul01d 9774	. 2 |- ( ph -> ( ( 1 / M ) x. 0 ) = 0 )
33	8, 28, 32	3eqtrd 2512	1 |- ( ph -> ( P ` Z ) = 0 )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   {crab 2818   C_ wss 3476   class class class wbr 4447   |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   Fincfn 7513   RRcr 9487   0cc0 9488   1c1 9489   x. cmul 9493   <_ cle 9625   / cdiv 10202   NNcn 10532   ZZ>=cuz 11078   ...cfz 11668   sum_csu 13467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468

This theorem is referenced by:  stoweidlem44  31344