Theorem stoweidlem37 37892

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 37885	. . 3 |- ( ( ph /\ Z e. T ) -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
8	1, 7	mpdan 673	. 2 |- ( ph -> ( P ` Z ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) )
9	5	fnvinran 37329	. . . . . . 7 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( G ` i ) e. Q )
10		fveq1 5862	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( h ` Z ) = ( ( G ` i ) ` Z ) )
11	10	eqeq1d 2452	. . . . . . . . 9 |- ( h = ( G ` i ) -> ( ( h ` Z ) = 0 <-> ( ( G ` i ) ` Z ) = 0 ) )
12		fveq1 5862	. . . . . . . . . . . 12 |- ( h = ( G ` i ) -> ( h ` t ) = ( ( G ` i ) ` t ) )
13	12	breq2d 4413	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( 0 <_ ( h ` t ) <-> 0 <_ ( ( G ` i ) ` t ) ) )
14	12	breq1d 4411	. . . . . . . . . . 11 |- ( h = ( G ` i ) -> ( ( h ` t ) <_ 1 <-> ( ( G ` i ) ` t ) <_ 1 ) )
15	13, 14	anbi12d 716	. . . . . . . . . 10 |- ( h = ( G ` i ) -> ( ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) <-> ( 0 <_ ( ( G ` i ) ` t ) /\ ( ( G ` i ) ` t ) <_ 1 ) ) )
16	15	ralbidv 2826	. . . . . . . . 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 716	. . . . . . . 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 3197	. . . . . . 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 200	. . . . . 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 764	. . . . 5 |- ( ( ph /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` Z ) = 0 )
21	20	sumeq2dv 13762	. . . 4 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = sum_ i e. ( 1 ... M ) 0 )
22		fzfi 12182	. . . . 5 |- ( 1 ... M ) e. Fin
23		olc 386	. . . . 5 |- ( ( 1 ... M ) e. Fin -> ( ( 1 ... M ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... M ) e. Fin ) )
24		sumz 13781	. . . . 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 2500	. . 3 |- ( ph -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) = 0 )
27	26	oveq2d 6304	. 2 |- ( ph -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` Z ) ) = ( ( 1 / M ) x. 0 ) )
28	4	nncnd 10622	. . . 4 |- ( ph -> M e. CC )
29	4	nnne0d 10651	. . . 4 |- ( ph -> M =/= 0 )
30	28, 29	reccld 10373	. . 3 |- ( ph -> ( 1 / M ) e. CC )
31	30	mul01d 9829	. 2 |- ( ph -> ( ( 1 / M ) x. 0 ) = 0 )
32	8, 27, 31	3eqtrd 2488	1 |- ( ph -> ( P ` Z ) = 0 )

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   {crab 2740   C_ wss 3403   class class class wbr 4401   |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   Fincfn 7566   RRcr 9535   0cc0 9536   1c1 9537   x. cmul 9541   <_ cle 9673   / cdiv 10266   NNcn 10606   ZZ>=cuz 11156   ...cfz 11781   sum_csu 13745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746

This theorem is referenced by:  stoweidlem44  37899