Theorem fzval2 9947

Description: An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
fzval2	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Proof of Theorem fzval2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzval 9946	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
2		zssre 9198	. . . . . . 7 |- ZZ C_ RR
3		ressxr 7942	. . . . . . 7 |- RR C_ RR*
4	2, 3	sstri 3151	. . . . . 6 |- ZZ C_ RR*
5	4	sseli 3138	. . . . 5 |- ( M e. ZZ -> M e. RR* )
6	4	sseli 3138	. . . . 5 |- ( N e. ZZ -> N e. RR* )
7		iccval 9856	. . . . 5 |- ( ( M e. RR* /\ N e. RR* ) -> ( M [,] N ) = { k e. RR* | ( M <_ k /\ k <_ N ) } )
8	5, 6, 7	syl2an 287	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M [,] N ) = { k e. RR* | ( M <_ k /\ k <_ N ) } )
9	8	ineq1d 3322	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M [,] N ) i^i ZZ ) = ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) )
10		inrab2 3395	. . . 4 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) }
11		sseqin2 3341	. . . . . 6 |- ( ZZ C_ RR* <-> ( RR* i^i ZZ ) = ZZ )
12	4, 11	mpbi 144	. . . . 5 |- ( RR* i^i ZZ ) = ZZ
13		rabeq 2718	. . . . 5 |- ( ( RR* i^i ZZ ) = ZZ -> { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
14	12, 13	ax-mp 5	. . . 4 |- { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) } = { k e. ZZ | ( M <_ k /\ k <_ N ) }
15	10, 14	eqtri 2186	. . 3 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ZZ | ( M <_ k /\ k <_ N ) }
16	9, 15	eqtr2di 2216	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = ( ( M [,] N ) i^i ZZ ) )
17	1, 16	eqtrd 2198	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {crab 2448   i^i cin 3115   C_ wss 3116   class class class wbr 3982  (class class class)co 5842   RRcr 7752   RR*cxr 7932   <_ cle 7934   ZZcz 9191   [,]cicc 9827   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-neg 8072  df-z 9192  df-icc 9831  df-fz 9945

This theorem is referenced by: (None)