Theorem mptfzshft 11828

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 2206	. 2 |- ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) = ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) )
2		elfzelz 10167	. . . 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 9520	. 2 |- ( ( ph /\ j e. ( ( M + K ) ... ( N + K ) ) ) -> ( j - K ) e. ZZ )
7		elfzelz 10167	. . . 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 9519	. 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 5972	. . . . . . 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 9397	. . . . . . . . 9 |- ( j e. ZZ -> j e. CC )
16		zcn 9397	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
17		npcan 8301	. . . . . . . . 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 2240	. . . . . 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 2284	. . . . 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 9520	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> ( j - K ) e. ZZ )
28	11, 27	eqeltrd 2283	. . . . . 6 |- ( ( ph /\ ( j e. ( ( M + K ) ... ( N + K ) ) /\ k = ( j - K ) ) ) -> k e. ZZ )
29		fzaddel 10201	. . . . . 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 1251	. . . . 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 1251	. . . . . 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 2283	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> j e. ( ( M + K ) ... ( N + K ) ) )
42	33	oveq1d 5972	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( k + K ) ) ) -> ( j - K ) = ( ( k + K ) - K ) )
43		zcn 9397	. . . . . . 7 |- ( k e. ZZ -> k e. CC )
44		pncan 8298	. . . . . . 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 2240	. . . 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 6162	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 1373   e. wcel 2177   |-> cmpt 4113   -1-1-onto->wf1o 5279  (class class class)co 5957   CCcc 7943   + caddc 7948   - cmin 8263   ZZcz 9392   ...cfz 10150

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-fz 10151

This theorem is referenced by:  fsumshft  11830  fprodshft  12004