Theorem mptfzshft 11588

Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (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		eqid 2193	. 2 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
2		elfzelz 10094	. . . 4 |- ( j e. ( ( M + K ) ... ( N + K ) ) -> j e. ZZ )
3	2	adantl 277	. . 3 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> j e. ZZ )
4		mptfzshft.1	. . . 4 |- ( ph -> K e. ZZ )
5	4	adantr 276	. . 3 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> K e. ZZ )
6	3, 5	zsubcld 9447	. 2 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> ( j - K ) e. ZZ )
7		elfzelz 10094	. . . 4 |- ( k e. ( M ... N ) -> k e. ZZ )
8	7	adantl 277	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> k e. ZZ )
9	4	adantr 276	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> K e. ZZ )
10	8, 9	zaddcld 9446	. 2 |- ( ( ph /\ k e. ( M ... N ) ) -> ( k + K ) e. ZZ )
11		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k = ( j - K ) )
12	11	oveq1d 5934	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k + K ) = ( ( j - K ) + K ) )
13	2	ad2antrl 490	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j e. ZZ )
14	4	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> K e. ZZ )
15		zcn 9325	. . . . . . . . 9 |- ( j e. ZZ -> j e. CC )
16		zcn 9325	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
17		npcan 8230	. . . . . . . . 9 |- ( ( j e. CC /\ K e. CC ) -> ( ( j - K ) + K ) = j )
18	15, 16, 17	syl2an 289	. . . . . . . 8 |- ( ( j e. ZZ /\ K e. ZZ ) -> ( ( j - K ) + K ) = j )
19	13, 14, 18	syl2anc 411	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( ( j - K ) + K ) = j )
20	12, 19	eqtr2d 2227	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j = ( k + K ) )
21		simprl 529	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
22	20, 21	eqeltrrd 2271	. . . . 5 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) )
23		mptfzshft.2	. . . . . . 7 |- ( ph -> M e. ZZ )
24	23	adantr 276	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> M e. ZZ )
25		mptfzshft.3	. . . . . . 7 |- ( ph -> N e. ZZ )
26	25	adantr 276	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> N e. ZZ )
27	13, 14	zsubcld 9447	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
28	11, 27	eqeltrd 2270	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
29		fzaddel 10128	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( k e. ZZ /\ K e. ZZ ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
30	24, 26, 28, 14, 29	syl22anc 1250	. . . . 5 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
31	22, 30	mpbird 167	. . . 4 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ( M ... N ) )
32	31, 20	jca 306	. . 3 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( k e. ( M ... N ) /\ j = ( k + K ) ) )
33		simprr 531	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j = ( k + K ) )
34		simprl 529	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k e. ( M ... N ) )
35	23	adantr 276	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> M e. ZZ )
36	25	adantr 276	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> N e. ZZ )
37	7	ad2antrl 490	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k e. ZZ )
38	4	adantr 276	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> K e. ZZ )
39	35, 36, 37, 38, 29	syl22anc 1250	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( k e. ( M ... N ) <-> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
40	34, 39	mpbid 147	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( k + K ) e. ( ( M + K ) ... ( N + K ) ) )
41	33, 40	eqeltrd 2270	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	33	oveq1d 5934	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 9325	. . . . . . 7 |- ( k e. ZZ -> k e. CC )
44		pncan 8227	. . . . . . 7 |- ( ( k e. CC /\ K e. CC ) -> ( ( k + K ) - K ) = k )
45	43, 16, 44	syl2an 289	. . . . . 6 |- ( ( k e. ZZ /\ K e. ZZ ) -> ( ( k + K ) - K ) = k )
46	37, 38, 45	syl2anc 411	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( ( k + K ) - K ) = k )
47	42, 46	eqtr2d 2227	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> k = ( j - K ) )
48	41, 47	jca 306	. . 3 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) )
49	32, 48	impbida 596	. 2 |- ( ph -> ( ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) <-> ( k e. ( M ... N ) /\ j = ( k + K ) ) ) )
50	1, 6, 10, 49	f1od 6123	1 |- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   |-> cmpt 4091   -1-1-onto->wf1o 5254  (class class class)co 5919   CCcc 7872   + caddc 7877   - cmin 8192   ZZcz 9320   ...cfz 10077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078

This theorem is referenced by:  fsumshft  11590  fprodshft  11764