Theorem fzval2 10345

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 10344	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
2		zssre 9584	. . . . . . 7 |- ZZ C_ RR
3		ressxr 8317	. . . . . . 7 |- RR C_ RR*
4	2, 3	sstri 3247	. . . . . 6 |- ZZ C_ RR*
5	4	sseli 3234	. . . . 5 |- ( M e. ZZ -> M e. RR* )
6	4	sseli 3234	. . . . 5 |- ( N e. ZZ -> N e. RR* )
7		iccval 10253	. . . . 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 3421	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M [,] N ) i^i ZZ ) = ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) )
10		inrab2 3494	. . . 4 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) }
11		sseqin2 3440	. . . . . 6 |- ( ZZ C_ RR* <-> ( RR* i^i ZZ ) = ZZ )
12	4, 11	mpbi 145	. . . . 5 |- ( RR* i^i ZZ ) = ZZ
13		rabeq 2805	. . . . 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 2253	. . 3 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ZZ | ( M <_ k /\ k <_ N ) }
16	9, 15	eqtr2di 2282	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = ( ( M [,] N ) i^i ZZ ) )
17	1, 16	eqtrd 2265	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {crab 2524   i^i cin 3210   C_ wss 3211   class class class wbr 4109  (class class class)co 6050   RRcr 8126   RR*cxr 8307   <_ cle 8309   ZZcz 9577   [,]cicc 10224   ...cfz 10342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-neg 8447  df-z 9578  df-icc 10228  df-fz 10343

This theorem is referenced by: (None)