Theorem fzval2 9968

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 9967	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
2		zssre 9219	. . . . . . 7 |- ZZ C_ RR
3		ressxr 7963	. . . . . . 7 |- RR C_ RR*
4	2, 3	sstri 3156	. . . . . 6 |- ZZ C_ RR*
5	4	sseli 3143	. . . . 5 |- ( M e. ZZ -> M e. RR* )
6	4	sseli 3143	. . . . 5 |- ( N e. ZZ -> N e. RR* )
7		iccval 9877	. . . . 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 3327	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M [,] N ) i^i ZZ ) = ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) )
10		inrab2 3400	. . . 4 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ( RR* i^i ZZ ) | ( M <_ k /\ k <_ N ) }
11		sseqin2 3346	. . . . . 6 |- ( ZZ C_ RR* <-> ( RR* i^i ZZ ) = ZZ )
12	4, 11	mpbi 144	. . . . 5 |- ( RR* i^i ZZ ) = ZZ
13		rabeq 2722	. . . . 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 2191	. . 3 |- ( { k e. RR* | ( M <_ k /\ k <_ N ) } i^i ZZ ) = { k e. ZZ | ( M <_ k /\ k <_ N ) }
16	9, 15	eqtr2di 2220	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> { k e. ZZ | ( M <_ k /\ k <_ N ) } = ( ( M [,] N ) i^i ZZ ) )
17	1, 16	eqtrd 2203	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {crab 2452   i^i cin 3120   C_ wss 3121   class class class wbr 3989  (class class class)co 5853   RRcr 7773   RR*cxr 7953   <_ cle 7955   ZZcz 9212   [,]cicc 9848   ...cfz 9965

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-neg 8093  df-z 9213  df-icc 9852  df-fz 9966

This theorem is referenced by: (None)