Theorem fourierdlem34 40358

Description: A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem34.p	|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem34.m	|- ( ph -> M e. NN )
fourierdlem34.q	|- ( ph -> Q e. ( P ` M ) )

Assertion

Ref	Expression
fourierdlem34	|- ( ph -> Q : ( 0 ... M ) -1-1-> RR )

Distinct variable groups:   A, m, p   B, m, p   i, M, m, p   Q, i, p   ph, i

Allowed substitution hints:   ph( m, p)   A( i)   B( i)   P( i, m, p)   Q( m)

Proof of Theorem fourierdlem34

Dummy variables k j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem34.q	. . . . 5 |- ( ph -> Q e. ( P ` M ) )
2		fourierdlem34.m	. . . . . 6 |- ( ph -> M e. NN )
3		fourierdlem34.p	. . . . . . 7 |- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
4	3	fourierdlem2 40326	. . . . . 6 |- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
5	2, 4	syl 17	. . . . 5 |- ( ph -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
6	1, 5	mpbid 222	. . . 4 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
7	6	simpld 475	. . 3 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
8		elmapi 7879	. . 3 |- ( Q e. ( RR ^m ( 0 ... M ) ) -> Q : ( 0 ... M ) --> RR )
9	7, 8	syl 17	. 2 |- ( ph -> Q : ( 0 ... M ) --> RR )
10		simplr 792	. . . . . 6 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) /\ -. i = j ) -> ( Q ` i ) = ( Q ` j ) )
11	9	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) e. RR )
12	11	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) e. RR )
13	9	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR )
14	13	ad4ant14 1293	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR )
15	14	adantllr 755	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR )
16		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( i = k -> ( i e. ( 0..^ M ) <-> k e. ( 0..^ M ) ) )
17	16	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( i = k -> ( ( ph /\ i e. ( 0..^ M ) ) <-> ( ph /\ k e. ( 0..^ M ) ) ) )
18		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( i = k -> ( Q ` i ) = ( Q ` k ) )
19		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( i = k -> ( i + 1 ) = ( k + 1 ) )
20	19	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( i = k -> ( Q ` ( i + 1 ) ) = ( Q ` ( k + 1 ) ) )
21	18, 20	breq12d 4666	. . . . . . . . . . . . . . . 16 |- ( i = k -> ( ( Q ` i ) < ( Q ` ( i + 1 ) ) <-> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) )
22	17, 21	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( i = k -> ( ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) <-> ( ( ph /\ k e. ( 0..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) ) )
23	6	simprrd 797	. . . . . . . . . . . . . . . 16 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
24	23	r19.21bi 2932	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
25	22, 24	chvarv 2263	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( 0..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) )
26	25	ad4ant14 1293	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) )
27	26	adantllr 755	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) )
28		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> i e. ( 0 ... M ) )
29		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> j e. ( 0 ... M ) )
30		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> i < j )
31	15, 27, 28, 29, 30	monoords 39511	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) < ( Q ` j ) )
32	12, 31	ltned 10173	. . . . . . . . . 10 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) =/= ( Q ` j ) )
33	32	neneqd 2799	. . . . . . . . 9 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> -. ( Q ` i ) = ( Q ` j ) )
34	33	adantlr 751	. . . . . . . 8 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ i < j ) -> -. ( Q ` i ) = ( Q ` j ) )
35		simpll 790	. . . . . . . . 9 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) )
36		elfzelz 12342	. . . . . . . . . . . 12 |- ( j e. ( 0 ... M ) -> j e. ZZ )
37	36	zred 11482	. . . . . . . . . . 11 |- ( j e. ( 0 ... M ) -> j e. RR )
38	37	ad3antlr 767	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j e. RR )
39		elfzelz 12342	. . . . . . . . . . . 12 |- ( i e. ( 0 ... M ) -> i e. ZZ )
40	39	zred 11482	. . . . . . . . . . 11 |- ( i e. ( 0 ... M ) -> i e. RR )
41	40	ad4antlr 769	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> i e. RR )
42		neqne 2802	. . . . . . . . . . . 12 |- ( -. i = j -> i =/= j )
43	42	necomd 2849	. . . . . . . . . . 11 |- ( -. i = j -> j =/= i )
44	43	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j =/= i )
45		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> -. i < j )
46	38, 41, 44, 45	lttri5d 39513	. . . . . . . . 9 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j < i )
47	9	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0 ... M ) ) -> ( Q ` j ) e. RR )
48	47	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) e. RR )
49	48	adantllr 755	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) e. RR )
50		simp-4l 806	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ... M ) ) -> ph )
51	50, 13	sylancom 701	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR )
52		simp-4l 806	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0..^ M ) ) -> ph )
53	52, 25	sylancom 701	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) )
54		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> j e. ( 0 ... M ) )
55		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> i e. ( 0 ... M ) )
56		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> j < i )
57	51, 53, 54, 55, 56	monoords 39511	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) < ( Q ` i ) )
58	49, 57	gtned 10172	. . . . . . . . . 10 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` i ) =/= ( Q ` j ) )
59	58	neneqd 2799	. . . . . . . . 9 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> -. ( Q ` i ) = ( Q ` j ) )
60	35, 46, 59	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> -. ( Q ` i ) = ( Q ` j ) )
61	34, 60	pm2.61dan 832	. . . . . . 7 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) -> -. ( Q ` i ) = ( Q ` j ) )
62	61	adantlr 751	. . . . . 6 |- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) /\ -. i = j ) -> -. ( Q ` i ) = ( Q ` j ) )
63	10, 62	condan 835	. . . . 5 |- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) -> i = j )
64	63	ex 450	. . . 4 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) -> ( ( Q ` i ) = ( Q ` j ) -> i = j ) )
65	64	ralrimiva 2966	. . 3 |- ( ( ph /\ i e. ( 0 ... M ) ) -> A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) )
66	65	ralrimiva 2966	. 2 |- ( ph -> A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) )
67		dff13 6512	. 2 |- ( Q : ( 0 ... M ) -1-1-> RR <-> ( Q : ( 0 ... M ) --> RR /\ A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) ) )
68	9, 66, 67	sylanbrc 698	1 |- ( ph -> Q : ( 0 ... M ) -1-1-> RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   ...cfz 12326  ..^cfzo 12465

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-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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466

This theorem is referenced by:  fourierdlem50  40373