Theorem axcontlem5 25848

Description: Lemma for axcont 25856. Compute the value of F. (Contributed by Scott Fenton, 18-Jun-2013.)

Hypotheses

Ref	Expression
axcontlem5.1	|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }
axcontlem5.2	|- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }

Assertion

Ref	Expression
axcontlem5	|- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) = T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )

Distinct variable groups:   t, D, x   i, p, t, x, N   P, i, t, x   x, T, i, t   U, i, p, t, x   i, Z, p, t, x

Allowed substitution hints:   D( i, p)   P( p)   T( p)   F( x, t, i, p)

Proof of Theorem axcontlem5

Step	Hyp	Ref	Expression
1		axcontlem5.1	. . . . . 6 |- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }
2		axcontlem5.2	. . . . . 6 |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }
3	1, 2	axcontlem2 25845	. . . . 5 |- ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) -> F : D -1-1-onto-> ( 0 [,) +oo ) )
4		f1of 6137	. . . . 5 |- ( F : D -1-1-onto-> ( 0 [,) +oo ) -> F : D --> ( 0 [,) +oo ) )
5	3, 4	syl 17	. . . 4 |- ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) -> F : D --> ( 0 [,) +oo ) )
6	5	ffvelrnda 6359	. . 3 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( F ` P ) e. ( 0 [,) +oo ) )
7		eleq1 2689	. . 3 |- ( ( F ` P ) = T -> ( ( F ` P ) e. ( 0 [,) +oo ) <-> T e. ( 0 [,) +oo ) ) )
8	6, 7	syl5ibcom 235	. 2 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) = T -> T e. ( 0 [,) +oo ) ) )
9		simpl 473	. . 3 |- ( ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) -> T e. ( 0 [,) +oo ) )
10	9	a1i 11	. 2 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) -> T e. ( 0 [,) +oo ) ) )
11		f1ofn 6138	. . . . . . 7 |- ( F : D -1-1-onto-> ( 0 [,) +oo ) -> F Fn D )
12	3, 11	syl 17	. . . . . 6 |- ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) -> F Fn D )
13		fnbrfvb 6236	. . . . . 6 |- ( ( F Fn D /\ P e. D ) -> ( ( F ` P ) = T <-> P F T ) )
14	12, 13	sylan 488	. . . . 5 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) = T <-> P F T ) )
15	14	3adant3 1081	. . . 4 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D /\ T e. ( 0 [,) +oo ) ) -> ( ( F ` P ) = T <-> P F T ) )
16		eleq1 2689	. . . . . . . 8 |- ( x = P -> ( x e. D <-> P e. D ) )
17		fveq1 6190	. . . . . . . . . . 11 |- ( x = P -> ( x ` i ) = ( P ` i ) )
18	17	eqeq1d 2624	. . . . . . . . . 10 |- ( x = P -> ( ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) )
19	18	ralbidv 2986	. . . . . . . . 9 |- ( x = P -> ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) )
20	19	anbi2d 740	. . . . . . . 8 |- ( x = P -> ( ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) <-> ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) )
21	16, 20	anbi12d 747	. . . . . . 7 |- ( x = P -> ( ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) <-> ( P e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) ) )
22		eleq1 2689	. . . . . . . . . 10 |- ( t = T -> ( t e. ( 0 [,) +oo ) <-> T e. ( 0 [,) +oo ) ) )
23		oveq2 6658	. . . . . . . . . . . . . 14 |- ( t = T -> ( 1 - t ) = ( 1 - T ) )
24	23	oveq1d 6665	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 1 - t ) x. ( Z ` i ) ) = ( ( 1 - T ) x. ( Z ` i ) ) )
25		oveq1 6657	. . . . . . . . . . . . 13 |- ( t = T -> ( t x. ( U ` i ) ) = ( T x. ( U ` i ) ) )
26	24, 25	oveq12d 6668	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) )
27	26	eqeq2d 2632	. . . . . . . . . . 11 |- ( t = T -> ( ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) )
28	27	ralbidv 2986	. . . . . . . . . 10 |- ( t = T -> ( A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) )
29	22, 28	anbi12d 747	. . . . . . . . 9 |- ( t = T -> ( ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
30	29	anbi2d 740	. . . . . . . 8 |- ( t = T -> ( ( P e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) <-> ( P e. D /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) ) )
31		anass 681	. . . . . . . . . . 11 |- ( ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ T e. ( 0 [,) +oo ) ) <-> ( P e. D /\ ( T e. ( 0 [,) +oo ) /\ T e. ( 0 [,) +oo ) ) ) )
32		anidm 676	. . . . . . . . . . . 12 |- ( ( T e. ( 0 [,) +oo ) /\ T e. ( 0 [,) +oo ) ) <-> T e. ( 0 [,) +oo ) )
33	32	anbi2i 730	. . . . . . . . . . 11 |- ( ( P e. D /\ ( T e. ( 0 [,) +oo ) /\ T e. ( 0 [,) +oo ) ) ) <-> ( P e. D /\ T e. ( 0 [,) +oo ) ) )
34	31, 33	bitr2i 265	. . . . . . . . . 10 |- ( ( P e. D /\ T e. ( 0 [,) +oo ) ) <-> ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ T e. ( 0 [,) +oo ) ) )
35	34	anbi1i 731	. . . . . . . . 9 |- ( ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) <-> ( ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ T e. ( 0 [,) +oo ) ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) )
36		anass 681	. . . . . . . . 9 |- ( ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) <-> ( P e. D /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
37		anass 681	. . . . . . . . 9 |- ( ( ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ T e. ( 0 [,) +oo ) ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) <-> ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
38	35, 36, 37	3bitr3i 290	. . . . . . . 8 |- ( ( P e. D /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) <-> ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
39	30, 38	syl6bb 276	. . . . . . 7 |- ( t = T -> ( ( P e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) <-> ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) ) )
40	21, 39, 2	brabg 4994	. . . . . 6 |- ( ( P e. D /\ T e. ( 0 [,) +oo ) ) -> ( P F T <-> ( ( P e. D /\ T e. ( 0 [,) +oo ) ) /\ ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) ) )
41	40	bianabs 924	. . . . 5 |- ( ( P e. D /\ T e. ( 0 [,) +oo ) ) -> ( P F T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
42	41	3adant1 1079	. . . 4 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D /\ T e. ( 0 [,) +oo ) ) -> ( P F T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
43	15, 42	bitrd 268	. . 3 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D /\ T e. ( 0 [,) +oo ) ) -> ( ( F ` P ) = T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )
44	43	3expia 1267	. 2 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( T e. ( 0 [,) +oo ) -> ( ( F ` P ) = T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) ) )
45	8, 10, 44	pm5.21ndd 369	1 |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) = T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   <.cop 4183   class class class wbr 4653   {copab 4712   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   - cmin 10266   NNcn 11020   [,)cico 12177   ...cfz 12326   EEcee 25768   Btwn cbtwn 25769

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-rmo 2920  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  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-div 10685  df-nn 11021  df-z 11378  df-uz 11688  df-ico 12181  df-icc 12182  df-fz 12327  df-ee 25771  df-btwn 25772

This theorem is referenced by:  axcontlem6  25849