Theorem stoweidlem23 27639

Description: This lemma is used to prove the existence of a function p_t as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that p_t ( t₀ ) = 0 , p_t ( t ) > 0, and 0 <= p_t <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem23.1	|- F/ t ph
stoweidlem23.2	|- F/_ t G
stoweidlem23.3	|- H = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) )
stoweidlem23.4	|- ( ( ph /\ f e. A ) -> f : T --> RR )
stoweidlem23.5	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem23.6	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweidlem23.7	|- ( ph -> S e. T )
stoweidlem23.8	|- ( ph -> Z e. T )
stoweidlem23.9	|- ( ph -> G e. A )
stoweidlem23.10	|- ( ph -> ( G ` S ) =/= ( G ` Z ) )

Assertion

Ref	Expression
stoweidlem23	|- ( ph -> ( H e. A /\ ( H ` S ) =/= ( H ` Z ) /\ ( H ` Z ) = 0 ) )

Distinct variable groups:   f, g, t, T   A, f, g   f, G, g   ph, f, g   g, Z, t   x, t, T   t, S   x, A   x, G   x, Z   ph, x

Allowed substitution hints:   ph( t)   A( t)   S( x, f, g)   G( t)   H( x, t, f, g)   Z( f)

Proof of Theorem stoweidlem23

Step	Hyp	Ref	Expression
1		stoweidlem23.3	. . 3 |- H = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) )
2		stoweidlem23.1	. . . . 5 |- F/ t ph
3		stoweidlem23.9	. . . . . . . . 9 |- ( ph -> G e. A )
4	3	ancli 535	. . . . . . . . 9 |- ( ph -> ( ph /\ G e. A ) )
5		eleq1 2464	. . . . . . . . . . . 12 |- ( f = G -> ( f e. A <-> G e. A ) )
6	5	anbi2d 685	. . . . . . . . . . 11 |- ( f = G -> ( ( ph /\ f e. A ) <-> ( ph /\ G e. A ) ) )
7		feq1 5535	. . . . . . . . . . 11 |- ( f = G -> ( f : T --> RR <-> G : T --> RR ) )
8	6, 7	imbi12d 312	. . . . . . . . . 10 |- ( f = G -> ( ( ( ph /\ f e. A ) -> f : T --> RR ) <-> ( ( ph /\ G e. A ) -> G : T --> RR ) ) )
9		stoweidlem23.4	. . . . . . . . . 10 |- ( ( ph /\ f e. A ) -> f : T --> RR )
10	8, 9	vtoclg 2971	. . . . . . . . 9 |- ( G e. A -> ( ( ph /\ G e. A ) -> G : T --> RR ) )
11	3, 4, 10	sylc 58	. . . . . . . 8 |- ( ph -> G : T --> RR )
12	11	fnvinran 27552	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. RR )
13	12	recnd 9070	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. CC )
14		stoweidlem23.8	. . . . . . . . 9 |- ( ph -> Z e. T )
15	11, 14	ffvelrnd 5830	. . . . . . . 8 |- ( ph -> ( G ` Z ) e. RR )
16	15	adantr 452	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. RR )
17	16	recnd 9070	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. CC )
18	13, 17	negsubd 9373	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + -u ( G ` Z ) ) = ( ( G ` t ) - ( G ` Z ) ) )
19	2, 18	mpteq2da 4254	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) )
20		simpr 448	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
21	15	renegcld 9420	. . . . . . . . 9 |- ( ph -> -u ( G ` Z ) e. RR )
22	21	adantr 452	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> -u ( G ` Z ) e. RR )
23		eqid 2404	. . . . . . . . 9 |- ( t e. T |-> -u ( G ` Z ) ) = ( t e. T |-> -u ( G ` Z ) )
24	23	fvmpt2 5771	. . . . . . . 8 |- ( ( t e. T /\ -u ( G ` Z ) e. RR ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
25	20, 22, 24	syl2anc 643	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
26	25	oveq2d 6056	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) = ( ( G ` t ) + -u ( G ` Z ) ) )
27	2, 26	mpteq2da 4254	. . . . 5 |- ( ph -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) = ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) )
28	21	ancli 535	. . . . . . 7 |- ( ph -> ( ph /\ -u ( G ` Z ) e. RR ) )
29		eleq1 2464	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( x e. RR <-> -u ( G ` Z ) e. RR ) )
30	29	anbi2d 685	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( ph /\ x e. RR ) <-> ( ph /\ -u ( G ` Z ) e. RR ) ) )
31		stoweidlem23.2	. . . . . . . . . . . . . 14 |- F/_ t G
32		nfcv 2540	. . . . . . . . . . . . . 14 |- F/_ t Z
33	31, 32	nffv 5694	. . . . . . . . . . . . 13 |- F/_ t ( G ` Z )
34	33	nfneg 9258	. . . . . . . . . . . 12 |- F/_ t -u ( G ` Z )
35	34	nfeq2 2551	. . . . . . . . . . 11 |- F/ t x = -u ( G ` Z )
36		simpl 444	. . . . . . . . . . 11 |- ( ( x = -u ( G ` Z ) /\ t e. T ) -> x = -u ( G ` Z ) )
37	35, 36	mpteq2da 4254	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( t e. T |-> x ) = ( t e. T |-> -u ( G ` Z ) ) )
38	37	eleq1d 2470	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> -u ( G ` Z ) ) e. A ) )
39	30, 38	imbi12d 312	. . . . . . . 8 |- ( x = -u ( G ` Z ) -> ( ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A ) <-> ( ( ph /\ -u ( G ` Z ) e. RR ) -> ( t e. T |-> -u ( G ` Z ) ) e. A ) ) )
40		stoweidlem23.6	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
41	39, 40	vtoclg 2971	. . . . . . 7 |- ( -u ( G ` Z ) e. RR -> ( ( ph /\ -u ( G ` Z ) e. RR ) -> ( t e. T |-> -u ( G ` Z ) ) e. A ) )
42	21, 28, 41	sylc 58	. . . . . 6 |- ( ph -> ( t e. T |-> -u ( G ` Z ) ) e. A )
43		stoweidlem23.5	. . . . . . 7 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
44		nfmpt1 4258	. . . . . . 7 |- F/_ t ( t e. T |-> -u ( G ` Z ) )
45	43, 31, 44	stoweidlem8 27624	. . . . . 6 |- ( ( ph /\ G e. A /\ ( t e. T |-> -u ( G ` Z ) ) e. A ) -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) e. A )
46	3, 42, 45	mpd3an23 1281	. . . . 5 |- ( ph -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) e. A )
47	27, 46	eqeltrrd 2479	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) e. A )
48	19, 47	eqeltrrd 2479	. . 3 |- ( ph -> ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) e. A )
49	1, 48	syl5eqel 2488	. 2 |- ( ph -> H e. A )
50		stoweidlem23.7	. . . . . 6 |- ( ph -> S e. T )
51	11, 50	ffvelrnd 5830	. . . . 5 |- ( ph -> ( G ` S ) e. RR )
52	51	recnd 9070	. . . 4 |- ( ph -> ( G ` S ) e. CC )
53	15	recnd 9070	. . . 4 |- ( ph -> ( G ` Z ) e. CC )
54		stoweidlem23.10	. . . 4 |- ( ph -> ( G ` S ) =/= ( G ` Z ) )
55	52, 53, 54	subne0d 9376	. . 3 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) =/= 0 )
56	51, 15	resubcld 9421	. . . 4 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) e. RR )
57		nfcv 2540	. . . . 5 |- F/_ t S
58	31, 57	nffv 5694	. . . . . 6 |- F/_ t ( G ` S )
59		nfcv 2540	. . . . . 6 |- F/_ t -
60	58, 59, 33	nfov 6063	. . . . 5 |- F/_ t ( ( G ` S ) - ( G ` Z ) )
61		fveq2 5687	. . . . . 6 |- ( t = S -> ( G ` t ) = ( G ` S ) )
62	61	oveq1d 6055	. . . . 5 |- ( t = S -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` S ) - ( G ` Z ) ) )
63	57, 60, 62, 1	fvmptf 5780	. . . 4 |- ( ( S e. T /\ ( ( G ` S ) - ( G ` Z ) ) e. RR ) -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
64	50, 56, 63	syl2anc 643	. . 3 |- ( ph -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
65	15, 15	resubcld 9421	. . . . 5 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) e. RR )
66	33, 59, 33	nfov 6063	. . . . . 6 |- F/_ t ( ( G ` Z ) - ( G ` Z ) )
67		fveq2 5687	. . . . . . 7 |- ( t = Z -> ( G ` t ) = ( G ` Z ) )
68	67	oveq1d 6055	. . . . . 6 |- ( t = Z -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` Z ) - ( G ` Z ) ) )
69	32, 66, 68, 1	fvmptf 5780	. . . . 5 |- ( ( Z e. T /\ ( ( G ` Z ) - ( G ` Z ) ) e. RR ) -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
70	14, 65, 69	syl2anc 643	. . . 4 |- ( ph -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
71	53	subidd 9355	. . . 4 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) = 0 )
72	70, 71	eqtrd 2436	. . 3 |- ( ph -> ( H ` Z ) = 0 )
73	55, 64, 72	3netr4d 2594	. 2 |- ( ph -> ( H ` S ) =/= ( H ` Z ) )
74	49, 73, 72	3jca 1134	1 |- ( ph -> ( H e. A /\ ( H ` S ) =/= ( H ` Z ) /\ ( H ` Z ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   F/wnf 1550   = wceq 1649   e. wcel 1721   F/_wnfc 2527   =/= wne 2567   e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   + caddc 8949   - cmin 9247   -ucneg 9248

This theorem is referenced by:  stoweidlem43  27659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250