Theorem stoweidlem37 37187

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 37180	. . 3 |- ( ( ph /\ Z e. T ) -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
8	1, 7	mpdan 666	. 2 |- ( ph -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
9	5	fnvinran 36769	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. Q )
10		fveq1 5848	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( h ` Z ) = ( ( G ` i ) ` Z ) )
11	10	eqeq1d 2404	. . . . . . . . 9 |- ( h = ( G ` i ) -> ( ( h ` Z ) = 0 <-> ( ( G ` i ) ` Z ) = 0 ) )
12		fveq1 5848	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( h ` t ) = ( ( G ` i ) ` t ) )
13	12	breq2d 4407	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( 0 <_ ( h ` t ) <-> 0 <_ ( ( G ` i ) ` t ) ) )
14	12	breq1d 4405	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( ( h ` t ) <_ 1 <-> ( ( G ` i ) ` t ) <_ 1 ) )
15	13, 14	anbi12d 709	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
16	15	ralbidv 2843	. . . . . . . . 9 |- ( 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 709	. . . . . . . 8 |- ( 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 3209	. . . . . . 7 |- ( ( 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	. . . . . 6 |- ( ( 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	simprld 757	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` Z ) = 0 )
21	20	sumeq2dv 13674	. . . 4 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = sum_ i e. ( 1 ... M ) 0 )
22		fzfi 12123	. . . . 5 |- ( 1 ... M ) e. Fin
23		olc 382	. . . . 5 |- ( ( 1 ... M ) e. Fin -> ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) )
24		sumz 13693	. . . . 5 |- ( ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) -> sum_ i e. ( 1 ... M ) 0 = 0 )
25	22, 23, 24	mp2b 10	. . . 4 |- sum_ i e. ( 1 ... M ) 0 = 0
26	21, 25	syl6eq 2459	. . 3 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = 0 )
27	26	oveq2d 6294	. 2 |- ( ph -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) = ( ( 1 / M ) x. 0 ) )
28	4	nncnd 10592	. . . 4 |- ( ph -> M e. CC )
29	4	nnne0d 10621	. . . 4 |- ( ph -> M =/= 0 )
30	28, 29	reccld 10354	. . 3 |- ( ph -> ( 1 / M ) e. CC )
31	30	mul01d 9813	. 2 |- ( ph -> ( ( 1 / M ) x. 0 ) = 0 )
32	8, 27, 31	3eqtrd 2447	1 |- ( ph -> ( P ` Z ) = 0 )

Syntax hints:   -> wi 4   \/ wo 366   /\ wa 367   = wceq 1405   e. wcel 1842   A.wral 2754   {crab 2758   C_ wss 3414   class class class wbr 4395   |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278   Fincfn 7554   RRcr 9521   0cc0 9522   1c1 9523   x. cmul 9527   <_ cle 9659   / cdiv 10247   NNcn 10576   ZZ>=cuz 11127   ...cfz 11726   sum_csu 13657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658

This theorem is referenced by:  stoweidlem44  37194