Theorem stoweidlem23 40240

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 574	. . . . . . . . 9 |- ( ph -> ( ph /\ G e. A ) )
5		eleq1 2689	. . . . . . . . . . . 12 |- ( f = G -> ( f e. A <-> G e. A ) )
6	5	anbi2d 740	. . . . . . . . . . 11 |- ( f = G -> ( ( ph /\ f e. A ) <-> ( ph /\ G e. A ) ) )
7		feq1 6026	. . . . . . . . . . 11 |- ( f = G -> ( f : T --> RR <-> G : T --> RR ) )
8	6, 7	imbi12d 334	. . . . . . . . . 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 3266	. . . . . . . . 9 |- ( G e. A -> ( ( ph /\ G e. A ) -> G : T --> RR ) )
11	3, 4, 10	sylc 65	. . . . . . . 8 |- ( ph -> G : T --> RR )
12	11	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. RR )
13	12	recnd 10068	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. CC )
14		stoweidlem23.8	. . . . . . . . 9 |- ( ph -> Z e. T )
15	11, 14	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( G ` Z ) e. RR )
16	15	adantr 481	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. RR )
17	16	recnd 10068	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. CC )
18	13, 17	negsubd 10398	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + -u ( G ` Z ) ) = ( ( G ` t ) - ( G ` Z ) ) )
19	2, 18	mpteq2da 4743	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) )
20		simpr 477	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
21	15	renegcld 10457	. . . . . . . . 9 |- ( ph -> -u ( G ` Z ) e. RR )
22	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> -u ( G ` Z ) e. RR )
23		eqid 2622	. . . . . . . . 9 |- ( t e. T |-> -u ( G ` Z ) ) = ( t e. T |-> -u ( G ` Z ) )
24	23	fvmpt2 6291	. . . . . . . 8 |- ( ( t e. T /\ -u ( G ` Z ) e. RR ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
25	20, 22, 24	syl2anc 693	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
26	25	oveq2d 6666	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) = ( ( G ` t ) + -u ( G ` Z ) ) )
27	2, 26	mpteq2da 4743	. . . . 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 574	. . . . . . 7 |- ( ph -> ( ph /\ -u ( G ` Z ) e. RR ) )
29		eleq1 2689	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( x e. RR <-> -u ( G ` Z ) e. RR ) )
30	29	anbi2d 740	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( ph /\ x e. RR ) <-> ( ph /\ -u ( G ` Z ) e. RR ) ) )
31		stoweidlem23.2	. . . . . . . . . . . . . 14 |- F/_ t G
32		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ t Z
33	31, 32	nffv 6198	. . . . . . . . . . . . 13 |- F/_ t ( G ` Z )
34	33	nfneg 10277	. . . . . . . . . . . 12 |- F/_ t -u ( G ` Z )
35	34	nfeq2 2780	. . . . . . . . . . 11 |- F/ t x = -u ( G ` Z )
36		simpl 473	. . . . . . . . . . 11 |- ( ( x = -u ( G ` Z ) /\ t e. T ) -> x = -u ( G ` Z ) )
37	35, 36	mpteq2da 4743	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( t e. T |-> x ) = ( t e. T |-> -u ( G ` Z ) ) )
38	37	eleq1d 2686	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> -u ( G ` Z ) ) e. A ) )
39	30, 38	imbi12d 334	. . . . . . . 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 3266	. . . . . . 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 65	. . . . . 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 4747	. . . . . . 7 |- F/_ t ( t e. T |-> -u ( G ` Z ) )
45	43, 31, 44	stoweidlem8 40225	. . . . . 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 1426	. . . . 5 |- ( ph -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) e. A )
47	27, 46	eqeltrrd 2702	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) e. A )
48	19, 47	eqeltrrd 2702	. . 3 |- ( ph -> ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) e. A )
49	1, 48	syl5eqel 2705	. 2 |- ( ph -> H e. A )
50		stoweidlem23.7	. . . . . 6 |- ( ph -> S e. T )
51	11, 50	ffvelrnd 6360	. . . . 5 |- ( ph -> ( G ` S ) e. RR )
52	51	recnd 10068	. . . 4 |- ( ph -> ( G ` S ) e. CC )
53	15	recnd 10068	. . . 4 |- ( ph -> ( G ` Z ) e. CC )
54		stoweidlem23.10	. . . 4 |- ( ph -> ( G ` S ) =/= ( G ` Z ) )
55	52, 53, 54	subne0d 10401	. . 3 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) =/= 0 )
56	51, 15	resubcld 10458	. . . 4 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) e. RR )
57		nfcv 2764	. . . . 5 |- F/_ t S
58	31, 57	nffv 6198	. . . . . 6 |- F/_ t ( G ` S )
59		nfcv 2764	. . . . . 6 |- F/_ t -
60	58, 59, 33	nfov 6676	. . . . 5 |- F/_ t ( ( G ` S ) - ( G ` Z ) )
61		fveq2 6191	. . . . . 6 |- ( t = S -> ( G ` t ) = ( G ` S ) )
62	61	oveq1d 6665	. . . . 5 |- ( t = S -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` S ) - ( G ` Z ) ) )
63	57, 60, 62, 1	fvmptf 6301	. . . 4 |- ( ( S e. T /\ ( ( G ` S ) - ( G ` Z ) ) e. RR ) -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
64	50, 56, 63	syl2anc 693	. . 3 |- ( ph -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
65	15, 15	resubcld 10458	. . . . 5 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) e. RR )
66	33, 59, 33	nfov 6676	. . . . . 6 |- F/_ t ( ( G ` Z ) - ( G ` Z ) )
67		fveq2 6191	. . . . . . 7 |- ( t = Z -> ( G ` t ) = ( G ` Z ) )
68	67	oveq1d 6665	. . . . . 6 |- ( t = Z -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` Z ) - ( G ` Z ) ) )
69	32, 66, 68, 1	fvmptf 6301	. . . . 5 |- ( ( Z e. T /\ ( ( G ` Z ) - ( G ` Z ) ) e. RR ) -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
70	14, 65, 69	syl2anc 693	. . . 4 |- ( ph -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
71	53	subidd 10380	. . . 4 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) = 0 )
72	70, 71	eqtrd 2656	. . 3 |- ( ph -> ( H ` Z ) = 0 )
73	55, 64, 72	3netr4d 2871	. 2 |- ( ph -> ( H ` S ) =/= ( H ` Z ) )
74	49, 73, 72	3jca 1242	1 |- ( ph -> ( H e. A /\ ( H ` S ) =/= ( H ` Z ) /\ ( H ` Z ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   - cmin 10266   -ucneg 10267

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-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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  stoweidlem43  40260