Theorem mptfzshft 11463

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 2187	. 2 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
2		elfzelz 10038	. . . 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 9393	. 2 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> ( j - K ) e. ZZ )
7		elfzelz 10038	. . . 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 9392	. 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 5903	. . . . . . 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 9271	. . . . . . . . 9 |- ( j e. ZZ -> j e. CC )
16		zcn 9271	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
17		npcan 8179	. . . . . . . . 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 2221	. . . . . 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 2265	. . . . 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 9393	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
28	11, 27	eqeltrd 2264	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
29		fzaddel 10072	. . . . . 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 1249	. . . . 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 1249	. . . . . 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 2264	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	33	oveq1d 5903	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 9271	. . . . . . 7 |- ( k e. ZZ -> k e. CC )
44		pncan 8176	. . . . . . 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 2221	. . . 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 6087	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 1363   e. wcel 2158   |-> cmpt 4076   -1-1-onto->wf1o 5227  (class class class)co 5888   CCcc 7822   + caddc 7827   - cmin 8141   ZZcz 9266   ...cfz 10021

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022

This theorem is referenced by:  fsumshft  11465  fprodshft  11639