Theorem fzval2 10393

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 10392	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
2		zssre 9630	. . . . . . 7 |- ZZ C_ RR
3		ressxr 8359	. . . . . . 7 |- RR C_ RR*
4	2, 3	sstri 3257	. . . . . 6 |- ZZ C_ RR*
5	4	sseli 3244	. . . . 5 |- ( M e. ZZ -> M e. RR* )
6	4	sseli 3244	. . . . 5 |- ( N e. ZZ -> N e. RR* )
7		iccval 10301	. . . . 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 3431	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M [,] N ) i^i ZZ ) = ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) )
10		inrab2 3506	. . . 4 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) }
11		sseqin2 3450	. . . . . 6 |- ( ZZ C_ RR* <-> ( RR* i^i ZZ ) = ZZ )
12	4, 11	mpbi 145	. . . . 5 |- ( RR* i^i ZZ ) = ZZ
13		rabeq 2813	. . . . 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 2259	. . 3 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ZZ | ( M <_ k /\ k <_ N ) }
16	9, 15	eqtr2di 2288	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = ( ( M [,] N ) i^i ZZ ) )
17	1, 16	eqtrd 2271	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   i^i cin 3219   C_ wss 3220   class class class wbr 4125  (class class class)co 6075   RRcr 8168   RR*cxr 8349   <_ cle 8351   ZZcz 9623   [,]cicc 10272   ...cfz 10390

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-neg 8490  df-z 9624  df-icc 10276  df-fz 10391

This theorem is referenced by: (None)