Theorem stoweidlem40 40257

Description: This lemma proves that q_n is in the subalgebra, as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent q_n in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem40.1	|- F/_ t P
stoweidlem40.2	|- F/ t ph
stoweidlem40.3	|- Q = ( t e. T |-> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ M ) )
stoweidlem40.4	|- F = ( t e. T |-> ( 1 - ( ( P ` t ) ^ N ) ) )
stoweidlem40.5	|- G = ( t e. T |-> 1 )
stoweidlem40.6	|- H = ( t e. T |-> ( ( P ` t ) ^ N ) )
stoweidlem40.7	|- ( ph -> P e. A )
stoweidlem40.8	|- ( ph -> P : T --> RR )
stoweidlem40.9	|- ( ( ph /\ f e. A ) -> f : T --> RR )
stoweidlem40.10	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem40.11	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweidlem40.12	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweidlem40.13	|- ( ph -> N e. NN )
stoweidlem40.14	|- ( ph -> M e. NN )

Assertion

Ref	Expression
stoweidlem40	|- ( ph -> Q e. A )

Distinct variable groups:   f, g, t, A   f, F, g   f, G, g   f, H, g   P, f, g   T, f, g, t   ph, f, g   x, t, A   t, M   t, N   x, T   ph, x

Allowed substitution hints:   ph( t)   P( x, t)   Q( x, t, f, g)   F( x, t)   G( x, t)   H( x, t)   M( x, f, g)   N( x, f, g)

Proof of Theorem stoweidlem40

Step	Hyp	Ref	Expression
1		stoweidlem40.3	. . 3 |- Q = ( t e. T |-> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ M ) )
2		stoweidlem40.2	. . . 4 |- F/ t ph
3		simpr 477	. . . . . . 7 |- ( ( ph /\ t e. T ) -> t e. T )
4		1red 10055	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> 1 e. RR )
5		stoweidlem40.8	. . . . . . . . . 10 |- ( ph -> P : T --> RR )
6	5	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> ( P ` t ) e. RR )
7		stoweidlem40.13	. . . . . . . . . . 11 |- ( ph -> N e. NN )
8	7	nnnn0d 11351	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
9	8	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. T ) -> N e. NN0 )
10	6, 9	reexpcld 13025	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) e. RR )
11	4, 10	resubcld 10458	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) e. RR )
12		stoweidlem40.4	. . . . . . . 8 |- F = ( t e. T |-> ( 1 - ( ( P ` t ) ^ N ) ) )
13	12	fvmpt2 6291	. . . . . . 7 |- ( ( t e. T /\ ( 1 - ( ( P ` t ) ^ N ) ) e. RR ) -> ( F ` t ) = ( 1 - ( ( P ` t ) ^ N ) ) )
14	3, 11, 13	syl2anc 693	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) = ( 1 - ( ( P ` t ) ^ N ) ) )
15	14	eqcomd 2628	. . . . 5 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) = ( F ` t ) )
16	15	oveq1d 6665	. . . 4 |- ( ( ph /\ t e. T ) -> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ M ) = ( ( F ` t ) ^ M ) )
17	2, 16	mpteq2da 4743	. . 3 |- ( ph -> ( t e. T |-> ( ( 1 - ( ( P ` t ) ^ N ) ) ^ M ) ) = ( t e. T |-> ( ( F ` t ) ^ M ) ) )
18	1, 17	syl5eq 2668	. 2 |- ( ph -> Q = ( t e. T |-> ( ( F ` t ) ^ M ) ) )
19		nfmpt1 4747	. . . 4 |- F/_ t ( t e. T |-> ( 1 - ( ( P ` t ) ^ N ) ) )
20	12, 19	nfcxfr 2762	. . 3 |- F/_ t F
21		stoweidlem40.9	. . 3 |- ( ( ph /\ f e. A ) -> f : T --> RR )
22		stoweidlem40.11	. . 3 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
23		stoweidlem40.12	. . 3 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
24		1re 10039	. . . . . . . . . 10 |- 1 e. RR
25		stoweidlem40.5	. . . . . . . . . . 11 |- G = ( t e. T |-> 1 )
26	25	fvmpt2 6291	. . . . . . . . . 10 |- ( ( t e. T /\ 1 e. RR ) -> ( G ` t ) = 1 )
27	24, 26	mpan2 707	. . . . . . . . 9 |- ( t e. T -> ( G ` t ) = 1 )
28	27	eqcomd 2628	. . . . . . . 8 |- ( t e. T -> 1 = ( G ` t ) )
29	28	adantl 482	. . . . . . 7 |- ( ( ph /\ t e. T ) -> 1 = ( G ` t ) )
30		stoweidlem40.6	. . . . . . . . . 10 |- H = ( t e. T |-> ( ( P ` t ) ^ N ) )
31	30	fvmpt2 6291	. . . . . . . . 9 |- ( ( t e. T /\ ( ( P ` t ) ^ N ) e. RR ) -> ( H ` t ) = ( ( P ` t ) ^ N ) )
32	3, 10, 31	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> ( H ` t ) = ( ( P ` t ) ^ N ) )
33	32	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( ( P ` t ) ^ N ) = ( H ` t ) )
34	29, 33	oveq12d 6668	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( 1 - ( ( P ` t ) ^ N ) ) = ( ( G ` t ) - ( H ` t ) ) )
35	2, 34	mpteq2da 4743	. . . . 5 |- ( ph -> ( t e. T |-> ( 1 - ( ( P ` t ) ^ N ) ) ) = ( t e. T |-> ( ( G ` t ) - ( H ` t ) ) ) )
36	12, 35	syl5eq 2668	. . . 4 |- ( ph -> F = ( t e. T |-> ( ( G ` t ) - ( H ` t ) ) ) )
37	23	stoweidlem4 40221	. . . . . . 7 |- ( ( ph /\ 1 e. RR ) -> ( t e. T |-> 1 ) e. A )
38	24, 37	mpan2 707	. . . . . 6 |- ( ph -> ( t e. T |-> 1 ) e. A )
39	25, 38	syl5eqel 2705	. . . . 5 |- ( ph -> G e. A )
40		stoweidlem40.1	. . . . . . 7 |- F/_ t P
41		stoweidlem40.7	. . . . . . 7 |- ( ph -> P e. A )
42	40, 2, 21, 22, 23, 41, 8	stoweidlem19 40236	. . . . . 6 |- ( ph -> ( t e. T |-> ( ( P ` t ) ^ N ) ) e. A )
43	30, 42	syl5eqel 2705	. . . . 5 |- ( ph -> H e. A )
44		nfmpt1 4747	. . . . . . 7 |- F/_ t ( t e. T |-> 1 )
45	25, 44	nfcxfr 2762	. . . . . 6 |- F/_ t G
46		nfmpt1 4747	. . . . . . 7 |- F/_ t ( t e. T |-> ( ( P ` t ) ^ N ) )
47	30, 46	nfcxfr 2762	. . . . . 6 |- F/_ t H
48		stoweidlem40.10	. . . . . 6 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
49	45, 47, 2, 21, 48, 22, 23	stoweidlem33 40250	. . . . 5 |- ( ( ph /\ G e. A /\ H e. A ) -> ( t e. T |-> ( ( G ` t ) - ( H ` t ) ) ) e. A )
50	39, 43, 49	mpd3an23 1426	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) - ( H ` t ) ) ) e. A )
51	36, 50	eqeltrd 2701	. . 3 |- ( ph -> F e. A )
52		stoweidlem40.14	. . . 4 |- ( ph -> M e. NN )
53	52	nnnn0d 11351	. . 3 |- ( ph -> M e. NN0 )
54	20, 2, 21, 22, 23, 51, 53	stoweidlem19 40236	. 2 |- ( ph -> ( t e. T |-> ( ( F ` t ) ^ M ) ) e. A )
55	18, 54	eqeltrd 2701	1 |- ( ph -> Q e. A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ^cexp 12860

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  stoweidlem45  40262