Theorem mptfzshft 11695

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 2204	. 2 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
2		elfzelz 10146	. . . 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 9499	. 2 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> ( j - K ) e. ZZ )
7		elfzelz 10146	. . . 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 9498	. 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 5958	. . . . . . 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 9376	. . . . . . . . 9 |- ( j e. ZZ -> j e. CC )
16		zcn 9376	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
17		npcan 8280	. . . . . . . . 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 2238	. . . . . 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 2282	. . . . 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 9499	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
28	11, 27	eqeltrd 2281	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
29		fzaddel 10180	. . . . . 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 2281	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	33	oveq1d 5958	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 9376	. . . . . . 7 |- ( k e. ZZ -> k e. CC )
44		pncan 8277	. . . . . . 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 2238	. . . 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 6148	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 1372   e. wcel 2175   |-> cmpt 4104   -1-1-onto->wf1o 5269  (class class class)co 5943   CCcc 7922   + caddc 7927   - cmin 8242   ZZcz 9371   ...cfz 10129

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-fz 10130

This theorem is referenced by:  fsumshft  11697  fprodshft  11871