Theorem stoweidlem37 40254

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 40247	. . 3 |- ( ( ph /\ Z e. T ) -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
8	1, 7	mpdan 702	. 2 |- ( ph -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
9	5	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. Q )
10		fveq1 6190	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( h ` Z ) = ( ( G ` i ) ` Z ) )
11	10	eqeq1d 2624	. . . . . . . . 9 |- ( h = ( G ` i ) -> ( ( h ` Z ) = 0 <-> ( ( G ` i ) ` Z ) = 0 ) )
12		fveq1 6190	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( h ` t ) = ( ( G ` i ) ` t ) )
13	12	breq2d 4665	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( 0 <_ ( h ` t ) <-> 0 <_ ( ( G ` i ) ` t ) ) )
14	12	breq1d 4663	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( ( h ` t ) <_ 1 <-> ( ( G ` i ) ` t ) <_ 1 ) )
15	13, 14	anbi12d 747	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
16	15	ralbidv 2986	. . . . . . . . 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 747	. . . . . . . 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 3366	. . . . . . 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 208	. . . . . 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 795	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` Z ) = 0 )
21	20	sumeq2dv 14433	. . . 4 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = sum_ i e. ( 1 ... M ) 0 )
22		fzfi 12771	. . . . 5 |- ( 1 ... M ) e. Fin
23		olc 399	. . . . 5 |- ( ( 1 ... M ) e. Fin -> ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) )
24		sumz 14453	. . . . 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 2672	. . 3 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = 0 )
27	26	oveq2d 6666	. 2 |- ( ph -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) = ( ( 1 / M ) x. 0 ) )
28	4	nncnd 11036	. . . 4 |- ( ph -> M e. CC )
29	4	nnne0d 11065	. . . 4 |- ( ph -> M =/= 0 )
30	28, 29	reccld 10794	. . 3 |- ( ph -> ( 1 / M ) e. CC )
31	30	mul01d 10235	. 2 |- ( ph -> ( ( 1 / M ) x. 0 ) = 0 )
32	8, 27, 31	3eqtrd 2660	1 |- ( ph -> ( P ` Z ) = 0 )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   / cdiv 10684   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  stoweidlem44  40261