Theorem iccpart 41352

Description: A special partition. Corresponds to fourierdlem2 40326 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)

Assertion

Ref	Expression
iccpart	|- ( M e. NN -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )

Distinct variable groups:   i, M   P, i

Proof of Theorem iccpart

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpval 41351	. . 3 |- ( M e. NN -> (RePart ` M ) = { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } )
2	1	eleq2d 2687	. 2 |- ( M e. NN -> ( P e. (RePart ` M ) <-> P e. { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } ) )
3		fveq1 6190	. . . . 5 |- ( p = P -> ( p ` i ) = ( P ` i ) )
4		fveq1 6190	. . . . 5 |- ( p = P -> ( p ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
5	3, 4	breq12d 4666	. . . 4 |- ( p = P -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( P ` i ) < ( P ` ( i + 1 ) ) ) )
6	5	ralbidv 2986	. . 3 |- ( p = P -> ( A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) )
7	6	elrab 3363	. 2 |- ( P e. { p e. ( RR* ^m ( 0 ... M ) ) | A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) } <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) )
8	2, 7	syl6bb 276	1 |- ( M e. NN -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   NNcn 11020   ...cfz 12326  ..^cfzo 12465  RePartciccp 41349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-iccp 41350

This theorem is referenced by:  iccpartimp  41353  iccpartres  41354  iccpartxr  41355  iccpartrn  41366  iccpartf  41367  iccpartnel  41374