Theorem stoweidlem30 37832

Description: This lemma is used to prove the existence of a function p as in Lemma 1 [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
stoweidlem30.1	|- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
stoweidlem30.2	|- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
stoweidlem30.3	|- ( ph -> M e. NN )
stoweidlem30.4	|- ( ph -> G : ( 1 ... M ) --> Q )
stoweidlem30.5	|- ( ( ph /\ f e. A ) -> f : T --> RR )

Assertion

Ref	Expression
stoweidlem30	|- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )

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, M, t   S, i

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

Proof of Theorem stoweidlem30

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2495	. . . . 5 |- ( s = S -> ( s e. T <-> S e. T ) )
2	1	anbi2d 708	. . . 4 |- ( s = S -> ( ( ph /\ s e. T ) <-> ( ph /\ S e. T ) ) )
3		fveq2 5882	. . . . 5 |- ( s = S -> ( P ` s ) = ( P ` S ) )
4		fveq2 5882	. . . . . . 7 |- ( s = S -> ( ( G ` i ) ` s ) = ( ( G ` i ) ` S ) )
5	4	sumeq2sdv 13770	. . . . . 6 |- ( s = S -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) )
6	5	oveq2d 6322	. . . . 5 |- ( s = S -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )
7	3, 6	eqeq12d 2444	. . . 4 |- ( s = S -> ( ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) <-> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
8	2, 7	imbi12d 321	. . 3 |- ( s = S -> ( ( ( ph /\ s e. T ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) ) <-> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) ) )
9		simpr 462	. . . 4 |- ( ( ph /\ s e. T ) -> s e. T )
10		stoweidlem30.3	. . . . . . 7 |- ( ph -> M e. NN )
11	10	nnrecred 10663	. . . . . 6 |- ( ph -> ( 1 / M ) e. RR )
12	11	adantr 466	. . . . 5 |- ( ( ph /\ s e. T ) -> ( 1 / M ) e. RR )
13		fzfid 12193	. . . . . 6 |- ( ( ph /\ s e. T ) -> ( 1 ... M ) e. Fin )
14		stoweidlem30.1	. . . . . . . . 9 |- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
15		stoweidlem30.4	. . . . . . . . 9 |- ( ph -> G : ( 1 ... M ) --> Q )
16		stoweidlem30.5	. . . . . . . . 9 |- ( ( ph /\ f e. A ) -> f : T --> RR )
17	14, 15, 16	stoweidlem15 37816	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( ( G ` i ) ` s ) e. RR /\ 0 <_ ( ( G ` i ) ` s ) /\ ( ( G ` i ) ` s ) <_ 1 ) )
18	17	simp1d 1017	. . . . . . 7 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ s e. T ) -> ( ( G ` i ) ` s ) e. RR )
19	18	an32s 811	. . . . . 6 |- ( ( ( ph /\ s e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` s ) e. RR )
20	13, 19	fsumrecl 13800	. . . . 5 |- ( ( ph /\ s e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) e. RR )
21	12, 20	remulcld 9679	. . . 4 |- ( ( ph /\ s e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR )
22		fveq2 5882	. . . . . . 7 |- ( t = s -> ( ( G ` i ) ` t ) = ( ( G ` i ) ` s ) )
23	22	sumeq2sdv 13770	. . . . . 6 |- ( t = s -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) = sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) )
24	23	oveq2d 6322	. . . . 5 |- ( t = s -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
25		stoweidlem30.2	. . . . 5 |- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
26	24, 25	fvmptg 5963	. . . 4 |- ( ( s e. T /\ ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) e. RR ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
27	9, 21, 26	syl2anc 665	. . 3 |- ( ( ph /\ s e. T ) -> ( P ` s ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` s ) ) )
28	8, 27	vtoclg 3139	. 2 |- ( S e. T -> ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) ) )
29	28	anabsi7 826	1 |- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1872   A.wral 2771   {crab 2775   class class class wbr 4423   |-> cmpt 4482   -->wf 5597   ` cfv 5601  (class class class)co 6306   RRcr 9546   0cc0 9547   1c1 9548   x. cmul 9552   <_ cle 9684   / cdiv 10277   NNcn 10617   ...cfz 11792   sum_csu 13752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-sup 7966  df-oi 8035  df-card 8382  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-n0 10878  df-z 10946  df-uz 11168  df-rp 11311  df-fz 11793  df-fzo 11924  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-clim 13552  df-sum 13753

This theorem is referenced by:  stoweidlem37  37839  stoweidlem38  37840  stoweidlem44  37846