Theorem mptfzshft 11607

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 2196	. 2 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
2		elfzelz 10100	. . . 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 9453	. 2 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> ( j - K ) e. ZZ )
7		elfzelz 10100	. . . 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 9452	. 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 5937	. . . . . . 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 9331	. . . . . . . . 9 |- ( j e. ZZ -> j e. CC )
16		zcn 9331	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
17		npcan 8235	. . . . . . . . 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 2230	. . . . . 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 2274	. . . . 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 9453	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
28	11, 27	eqeltrd 2273	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
29		fzaddel 10134	. . . . . 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 2273	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	33	oveq1d 5937	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 9331	. . . . . . 7 |- ( k e. ZZ -> k e. CC )
44		pncan 8232	. . . . . . 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 2230	. . . 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 6126	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 2167   |-> cmpt 4094   -1-1-onto->wf1o 5257  (class class class)co 5922   CCcc 7877   + caddc 7882   - cmin 8197   ZZcz 9326   ...cfz 10083

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084

This theorem is referenced by:  fsumshft  11609  fprodshft  11783