Theorem fzval2 10133

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 10132	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
2		zssre 9379	. . . . . . 7 |- ZZ C_ RR
3		ressxr 8116	. . . . . . 7 |- RR C_ RR*
4	2, 3	sstri 3202	. . . . . 6 |- ZZ C_ RR*
5	4	sseli 3189	. . . . 5 |- ( M e. ZZ -> M e. RR* )
6	4	sseli 3189	. . . . 5 |- ( N e. ZZ -> N e. RR* )
7		iccval 10042	. . . . 5 |- ( ( M e. RR* /\ N e. RR* ) -> ( M [,] N ) = { k e. RR* | ( M <_ k /\ k <_ N ) } )
8	5, 6, 7	syl2an 289	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M [,] N ) = { k e. RR* | ( M <_ k /\ k <_ N ) } )
9	8	ineq1d 3373	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M [,] N ) i^i ZZ ) = ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) )
10		inrab2 3446	. . . 4 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) }
11		sseqin2 3392	. . . . . 6 |- ( ZZ C_ RR* <-> ( RR* i^i ZZ ) = ZZ )
12	4, 11	mpbi 145	. . . . 5 |- ( RR* i^i ZZ ) = ZZ
13		rabeq 2764	. . . . 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 2226	. . 3 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ZZ | ( M <_ k /\ k <_ N ) }
16	9, 15	eqtr2di 2255	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = ( ( M [,] N ) i^i ZZ ) )
17	1, 16	eqtrd 2238	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {crab 2488   i^i cin 3165   C_ wss 3166   class class class wbr 4044  (class class class)co 5944   RRcr 7924   RR*cxr 8106   <_ cle 8108   ZZcz 9372   [,]cicc 10013   ...cfz 10130

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-neg 8246  df-z 9373  df-icc 10017  df-fz 10131

This theorem is referenced by: (None)