Theorem stoweidlem23 31351

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 551	. . . . . . . . 9 |- ( ph -> ( ph /\ G e. A ) )
5		eleq1 2539	. . . . . . . . . . . 12 |- ( f = G -> ( f e. A <-> G e. A ) )
6	5	anbi2d 703	. . . . . . . . . . 11 |- ( f = G -> ( ( ph /\ f e. A ) <-> ( ph /\ G e. A ) ) )
7		feq1 5713	. . . . . . . . . . 11 |- ( f = G -> ( f : T --> RR <-> G : T --> RR ) )
8	6, 7	imbi12d 320	. . . . . . . . . 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 3171	. . . . . . . . 9 |- ( G e. A -> ( ( ph /\ G e. A ) -> G : T --> RR ) )
11	3, 4, 10	sylc 60	. . . . . . . 8 |- ( ph -> G : T --> RR )
12	11	fnvinran 30995	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. RR )
13	12	recnd 9622	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` t ) e. CC )
14		stoweidlem23.8	. . . . . . . . 9 |- ( ph -> Z e. T )
15	11, 14	ffvelrnd 6022	. . . . . . . 8 |- ( ph -> ( G ` Z ) e. RR )
16	15	adantr 465	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. RR )
17	16	recnd 9622	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( G ` Z ) e. CC )
18	13, 17	negsubd 9936	. . . . 5 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + -u ( G ` Z ) ) = ( ( G ` t ) - ( G ` Z ) ) )
19	2, 18	mpteq2da 4532	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) = ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) )
20		simpr 461	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> t e. T )
21	15	renegcld 9986	. . . . . . . . 9 |- ( ph -> -u ( G ` Z ) e. RR )
22	21	adantr 465	. . . . . . . 8 |- ( ( ph /\ t e. T ) -> -u ( G ` Z ) e. RR )
23		eqid 2467	. . . . . . . . 9 |- ( t e. T |-> -u ( G ` Z ) ) = ( t e. T |-> -u ( G ` Z ) )
24	23	fvmpt2 5957	. . . . . . . 8 |- ( ( t e. T /\ -u ( G ` Z ) e. RR ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
25	20, 22, 24	syl2anc 661	. . . . . . 7 |- ( ( ph /\ t e. T ) -> ( ( t e. T |-> -u ( G ` Z ) ) ` t ) = -u ( G ` Z ) )
26	25	oveq2d 6300	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) = ( ( G ` t ) + -u ( G ` Z ) ) )
27	2, 26	mpteq2da 4532	. . . . 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 551	. . . . . . 7 |- ( ph -> ( ph /\ -u ( G ` Z ) e. RR ) )
29		eleq1 2539	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( x e. RR <-> -u ( G ` Z ) e. RR ) )
30	29	anbi2d 703	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( ph /\ x e. RR ) <-> ( ph /\ -u ( G ` Z ) e. RR ) ) )
31		stoweidlem23.2	. . . . . . . . . . . . . 14 |- F/_ t G
32		nfcv 2629	. . . . . . . . . . . . . 14 |- F/_ t Z
33	31, 32	nffv 5873	. . . . . . . . . . . . 13 |- F/_ t ( G ` Z )
34	33	nfneg 9816	. . . . . . . . . . . 12 |- F/_ t -u ( G ` Z )
35	34	nfeq2 2646	. . . . . . . . . . 11 |- F/ t x = -u ( G ` Z )
36		simpl 457	. . . . . . . . . . 11 |- ( ( x = -u ( G ` Z ) /\ t e. T ) -> x = -u ( G ` Z ) )
37	35, 36	mpteq2da 4532	. . . . . . . . . 10 |- ( x = -u ( G ` Z ) -> ( t e. T |-> x ) = ( t e. T |-> -u ( G ` Z ) ) )
38	37	eleq1d 2536	. . . . . . . . 9 |- ( x = -u ( G ` Z ) -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> -u ( G ` Z ) ) e. A ) )
39	30, 38	imbi12d 320	. . . . . . . 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 3171	. . . . . . 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 60	. . . . . 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 4536	. . . . . . 7 |- F/_ t ( t e. T |-> -u ( G ` Z ) )
45	43, 31, 44	stoweidlem8 31336	. . . . . 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 1326	. . . . 5 |- ( ph -> ( t e. T |-> ( ( G ` t ) + ( ( t e. T |-> -u ( G ` Z ) ) ` t ) ) ) e. A )
47	27, 46	eqeltrrd 2556	. . . 4 |- ( ph -> ( t e. T |-> ( ( G ` t ) + -u ( G ` Z ) ) ) e. A )
48	19, 47	eqeltrrd 2556	. . 3 |- ( ph -> ( t e. T |-> ( ( G ` t ) - ( G ` Z ) ) ) e. A )
49	1, 48	syl5eqel 2559	. 2 |- ( ph -> H e. A )
50		stoweidlem23.7	. . . . . 6 |- ( ph -> S e. T )
51	11, 50	ffvelrnd 6022	. . . . 5 |- ( ph -> ( G ` S ) e. RR )
52	51	recnd 9622	. . . 4 |- ( ph -> ( G ` S ) e. CC )
53	15	recnd 9622	. . . 4 |- ( ph -> ( G ` Z ) e. CC )
54		stoweidlem23.10	. . . 4 |- ( ph -> ( G ` S ) =/= ( G ` Z ) )
55	52, 53, 54	subne0d 9939	. . 3 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) =/= 0 )
56	51, 15	resubcld 9987	. . . 4 |- ( ph -> ( ( G ` S ) - ( G ` Z ) ) e. RR )
57		nfcv 2629	. . . . 5 |- F/_ t S
58	31, 57	nffv 5873	. . . . . 6 |- F/_ t ( G ` S )
59		nfcv 2629	. . . . . 6 |- F/_ t -
60	58, 59, 33	nfov 6307	. . . . 5 |- F/_ t ( ( G ` S ) - ( G ` Z ) )
61		fveq2 5866	. . . . . 6 |- ( t = S -> ( G ` t ) = ( G ` S ) )
62	61	oveq1d 6299	. . . . 5 |- ( t = S -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` S ) - ( G ` Z ) ) )
63	57, 60, 62, 1	fvmptf 5966	. . . 4 |- ( ( S e. T /\ ( ( G ` S ) - ( G ` Z ) ) e. RR ) -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
64	50, 56, 63	syl2anc 661	. . 3 |- ( ph -> ( H ` S ) = ( ( G ` S ) - ( G ` Z ) ) )
65	15, 15	resubcld 9987	. . . . 5 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) e. RR )
66	33, 59, 33	nfov 6307	. . . . . 6 |- F/_ t ( ( G ` Z ) - ( G ` Z ) )
67		fveq2 5866	. . . . . . 7 |- ( t = Z -> ( G ` t ) = ( G ` Z ) )
68	67	oveq1d 6299	. . . . . 6 |- ( t = Z -> ( ( G ` t ) - ( G ` Z ) ) = ( ( G ` Z ) - ( G ` Z ) ) )
69	32, 66, 68, 1	fvmptf 5966	. . . . 5 |- ( ( Z e. T /\ ( ( G ` Z ) - ( G ` Z ) ) e. RR ) -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
70	14, 65, 69	syl2anc 661	. . . 4 |- ( ph -> ( H ` Z ) = ( ( G ` Z ) - ( G ` Z ) ) )
71	53	subidd 9918	. . . 4 |- ( ph -> ( ( G ` Z ) - ( G ` Z ) ) = 0 )
72	70, 71	eqtrd 2508	. . 3 |- ( ph -> ( H ` Z ) = 0 )
73	55, 64, 72	3netr4d 2772	. 2 |- ( ph -> ( H ` S ) =/= ( H ` Z ) )
74	49, 73, 72	3jca 1176	1 |- ( ph -> ( H e. A /\ ( H ` S ) =/= ( H ` Z ) /\ ( H ` Z ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   F/wnf 1599   e. wcel 1767   F/_wnfc 2615   =/= wne 2662   |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   + caddc 9495   - cmin 9805   -ucneg 9806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-neg 9808

This theorem is referenced by:  stoweidlem43  31371