Theorem mptfzshft 14510

Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 14512. (Contributed by AV, 24-Aug-2019.)

Hypotheses

Ref	Expression
mptfzshft.1	|- ( ph -> K e. ZZ )
mptfzshft.2	|- ( ph -> M e. ZZ )
mptfzshft.3	|- ( ph -> N e. ZZ )

Assertion

Ref	Expression
mptfzshft	|- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )

Distinct variable groups:   j, K   j, M   j, N   ph, j

Proof of Theorem mptfzshft

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( j - K ) e. _V
2		eqid 2622	. . . 4 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
3	1, 2	fnmpti 6022	. . 3 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( ( M + K ) ... ( N + K ) )
4	3	a1i 11	. 2 |- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( ( M + K ) ... ( N + K ) ) )
5		ovex 6678	. . . 4 |- ( k + K ) e. _V
6		eqid 2622	. . . 4 |- ( k e. ( M ... N ) |-> ( k + K ) ) = ( k e. ( M ... N ) |-> ( k + K ) )
7	5, 6	fnmpti 6022	. . 3 |- ( k e. ( M ... N ) |-> ( k + K ) ) Fn ( M ... N )
8		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k = ( j - K ) )
9	8	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k + K ) = ( ( j - K ) + K ) )
10		elfzelz 12342	. . . . . . . . . . . 12 |- ( j e. ( ( M + K ) ... ( N + K ) ) -> j e. ZZ )
11	10	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j e. ZZ )
12		mptfzshft.1	. . . . . . . . . . . 12 |- ( ph -> K e. ZZ )
13	12	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> K e. ZZ )
14		zcn 11382	. . . . . . . . . . . 12 |- ( j e. ZZ -> j e. CC )
15		zcn 11382	. . . . . . . . . . . 12 |- ( K e. ZZ -> K e. CC )
16		npcan 10290	. . . . . . . . . . . 12 |- ( ( j e. CC /\ K e. CC ) -> ( ( j - K ) + K ) = j )
17	14, 15, 16	syl2an 494	. . . . . . . . . . 11 |- ( ( j e. ZZ /\ K e. ZZ ) -> ( ( j - K ) + K ) = j )
18	11, 13, 17	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( ( j - K ) + K ) = j )
19	9, 18	eqtr2d 2657	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j = ( k + K ) )
20		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
21	19, 20	eqeltrrd 2702	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) )
22		mptfzshft.2	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> M e. ZZ )
24		mptfzshft.3	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> N e. ZZ )
26	11, 13	zsubcld 11487	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
27	8, 26	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
28		fzaddel 12375	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( k e. ZZ /\ K e. ZZ ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
29	23, 25, 27, 13, 28	syl22anc 1327	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
30	21, 29	mpbird 247	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ( M ... N ) )
31	30, 19	jca 554	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k e. ( M ... N ) /\ j = ( k + K ) ) )
32		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j = ( k + K ) )
33		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k e. ( M ... N ) )
34	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> M e. ZZ )
35	24	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> N e. ZZ )
36		elfzelz 12342	. . . . . . . . . . 11 |- ( k e. ( M ... N ) -> k e. ZZ )
37	36	ad2antrl 764	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k e. ZZ )
38	12	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> K e. ZZ )
39	34, 35, 37, 38, 28	syl22anc 1327	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
40	33, 39	mpbid 222	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) )
41	32, 40	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	32	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 11382	. . . . . . . . . 10 |- ( k e. ZZ -> k e. CC )
44		pncan 10287	. . . . . . . . . 10 |- ( ( k e. CC /\ K e. CC ) -> ( ( k + K ) - K ) = k )
45	43, 15, 44	syl2an 494	. . . . . . . . 9 |- ( ( k e. ZZ /\ K e. ZZ ) -> ( ( k + K ) - K ) = k )
46	37, 38, 45	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( ( k + K ) - K ) = k )
47	42, 46	eqtr2d 2657	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k = ( j - K ) )
48	41, 47	jca 554	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) )
49	31, 48	impbida 877	. . . . 5 |- ( ph -> ( ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) <-> ( k e. ( M ... N ) /\ j = ( k + K ) ) ) )
50	49	mptcnv 5534	. . . 4 |- ( ph -> `' ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( k e. ( M ... N ) |-> ( k + K ) ) )
51	50	fneq1d 5981	. . 3 |- ( ph -> ( `' ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( M ... N ) <-> ( k e. ( M ... N ) |-> ( k + K ) ) Fn ( M ... N ) ) )
52	7, 51	mpbiri 248	. 2 |- ( ph -> `' ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( M ... N ) )
53		dff1o4 6145	. 2 |- ( ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) <-> ( ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( ( M + K ) ... ( N + K ) ) /\ `' ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) Fn ( M ... N ) ) )
54	4, 52, 53	sylanbrc 698	1 |- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   ZZcz 11377   ...cfz 12326

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-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

This theorem is referenced by:  fsumshft  14512  fprodshft  14706  gsummptshft  18336