Theorem stoweidlem23 37747

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 554	. . . . . . . . 9 |- ( ph -> ( ph /\ G e. A ) )
5		eleq1 2495	. . . . . . . . . . . 12 |- ( f = G -> ( f e. A <-> G e. A ) )
6	5	anbi2d 709	. . . . . . . . . . 11 |- ( f = G -> ( ( ph /\ f e. A ) <-> ( ph /\ G e. A ) ) )
7		feq1 5726	. . . . . . . . . . 11 |- ( f = G -> ( f : T --> RR <-> G : T --> RR ) )
8	6, 7	imbi12d 322	. . . . . . . . . 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 3140	. . . . . . . . 9 |- ( G e. A -> ( ( ph /\ G e. A ) -> G : T --> RR ) )
11	3, 4, 10	sylc 63	. . . . . . . 8 |- ( ph -> G : T --> RR )
12	11	fnvinran 37240	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. RR )
13	12	recnd 9671	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. CC )
14		stoweidlem23.8	. . . . . . . . 9 |- ( ph -> Z e. T )
15	11, 14	ffvelrnd 6036	. . . . . . . 8 |- ( ph -> ( G ` Z ) e. RR )
16	15	adantr 467	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. RR )
17	16	recnd 9671	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. CC )
18	13, 17	negsubd 9994	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + -u ( G ` Z ) ) = ( ( G ` t ) - ( G ` Z ) ) )
19	2, 18	mpteq2da 4507	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) )
20		simpr 463	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
21	15	renegcld 10048	. . . . . . . . 9 |- ( ph -> -u ( G ` Z ) e. RR )
22	21	adantr 467	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> -u ( G ` Z ) e. RR )
23		eqid 2423	. . . . . . . . 9 |- ( t e. T |-> -u ( G ` Z ) ) = ( t e. T |-> -u ( G ` Z ) )
24	23	fvmpt2 5971	. . . . . . . 8 |- ( ( t e. T /\ -u ( G ` Z ) e. RR ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
25	20, 22, 24	syl2anc 666	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
26	25	oveq2d 6319	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) = ( ( G ` t ) + -u ( G ` Z ) ) )
27	2, 26	mpteq2da 4507	. . . . 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 554	. . . . . . 7 |- ( ph -> ( ph /\ -u ( G ` Z ) e. RR ) )
29		eleq1 2495	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( x e. RR <-> -u ( G ` Z ) e. RR ) )
30	29	anbi2d 709	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( ph /\ x e. RR ) <-> ( ph /\ -u ( G ` Z ) e. RR ) ) )
31		stoweidlem23.2	. . . . . . . . . . . . . 14 |- F/_ t G
32		nfcv 2585	. . . . . . . . . . . . . 14 |- F/_ t Z
33	31, 32	nffv 5886	. . . . . . . . . . . . 13 |- F/_ t ( G ` Z )
34	33	nfneg 9873	. . . . . . . . . . . 12 |- F/_ t -u ( G ` Z )
35	34	nfeq2 2602	. . . . . . . . . . 11 |- F/ t x = -u ( G ` Z )
36		simpl 459	. . . . . . . . . . 11 |- ( ( x = -u ( G ` Z ) /\ t e. T ) -> x = -u ( G ` Z ) )
37	35, 36	mpteq2da 4507	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( t e. T |-> x ) = ( t e. T |-> -u ( G ` Z ) ) )
38	37	eleq1d 2492	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> -u ( G ` Z ) ) e. A ) )
39	30, 38	imbi12d 322	. . . . . . . 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 3140	. . . . . . 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 63	. . . . . 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 4511	. . . . . . 7 |- F/_ t ( t e. T |-> -u ( G ` Z ) )
45	43, 31, 44	stoweidlem8 37732	. . . . . 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 1363	. . . . 5 |- ( ph -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) e. A )
47	27, 46	eqeltrrd 2512	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) e. A )
48	19, 47	eqeltrrd 2512	. . 3 |- ( ph -> ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) e. A )
49	1, 48	syl5eqel 2515	. 2 |- ( ph -> H e. A )
50		stoweidlem23.7	. . . . . 6 |- ( ph -> S e. T )
51	11, 50	ffvelrnd 6036	. . . . 5 |- ( ph -> ( G ` S ) e. RR )
52	51	recnd 9671	. . . 4 |- ( ph -> ( G ` S ) e. CC )
53	15	recnd 9671	. . . 4 |- ( ph -> ( G ` Z ) e. CC )
54		stoweidlem23.10	. . . 4 |- ( ph -> ( G ` S ) =/= ( G ` Z ) )
55	52, 53, 54	subne0d 9997	. . 3 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) =/= 0 )
56	51, 15	resubcld 10049	. . . 4 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) e. RR )
57		nfcv 2585	. . . . 5 |- F/_ t S
58	31, 57	nffv 5886	. . . . . 6 |- F/_ t ( G ` S )
59		nfcv 2585	. . . . . 6 |- F/_ t -
60	58, 59, 33	nfov 6329	. . . . 5 |- F/_ t ( ( G ` S ) - ( G ` Z ) )
61		fveq2 5879	. . . . . 6 |- ( t = S -> ( G ` t ) = ( G ` S ) )
62	61	oveq1d 6318	. . . . 5 |- ( t = S -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` S ) - ( G ` Z ) ) )
63	57, 60, 62, 1	fvmptf 5980	. . . 4 |- ( ( S e. T /\ ( ( G ` S ) - ( G ` Z ) ) e. RR ) -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
64	50, 56, 63	syl2anc 666	. . 3 |- ( ph -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
65	15, 15	resubcld 10049	. . . . 5 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) e. RR )
66	33, 59, 33	nfov 6329	. . . . . 6 |- F/_ t ( ( G ` Z ) - ( G ` Z ) )
67		fveq2 5879	. . . . . . 7 |- ( t = Z -> ( G ` t ) = ( G ` Z ) )
68	67	oveq1d 6318	. . . . . 6 |- ( t = Z -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` Z ) - ( G ` Z ) ) )
69	32, 66, 68, 1	fvmptf 5980	. . . . 5 |- ( ( Z e. T /\ ( ( G ` Z ) - ( G ` Z ) ) e. RR ) -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
70	14, 65, 69	syl2anc 666	. . . 4 |- ( ph -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
71	53	subidd 9976	. . . 4 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) = 0 )
72	70, 71	eqtrd 2464	. . 3 |- ( ph -> ( H ` Z ) = 0 )
73	55, 64, 72	3netr4d 2730	. 2 |- ( ph -> ( H ` S ) =/= ( H ` Z ) )
74	49, 73, 72	3jca 1186	1 |- ( ph -> ( H e. A /\ ( H ` S ) =/= ( H ` Z ) /\ ( H ` Z ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 983   = wceq 1438   F/wnf 1664   e. wcel 1869   F/_wnfc 2571   =/= wne 2619   |-> cmpt 4480   -->wf 5595   ` cfv 5599  (class class class)co 6303   RRcr 9540   0cc0 9541   + caddc 9544   - cmin 9862   -ucneg 9863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-ltxr 9682  df-sub 9864  df-neg 9865

This theorem is referenced by:  stoweidlem43  37768