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Theorem iccpartimp 41353
Description: Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
Assertion
Ref Expression
iccpartimp |- ( ( M e. NN /\ P e. (RePart ` M ) /\ I e. ( 0..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` I ) < ( P ` ( I + 1 ) ) ) )
Proof of Theorem iccpartimp
Dummy variable i is distinct from all other variables.
StepHypRef Expression
1 iccpart 41352 . . 3 |- ( M e. NN -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )
2 fveq2 6191 . . . . . . . 8 |- ( i = I -> ( P ` i ) = ( P ` I ) )
3 oveq1 6657 . . . . . . . . 9 |- ( i = I -> ( i + 1 ) = ( I + 1 ) )
43fveq2d 6195 . . . . . . . 8 |- ( i = I -> ( P ` ( i + 1 ) ) = ( P ` ( I + 1 ) ) )
52, 4breq12d 4666 . . . . . . 7 |- ( i = I -> ( ( P ` i ) < ( P ` ( i + 1 ) ) <-> ( P ` I ) < ( P ` ( I + 1 ) ) ) )
65rspcva 3307 . . . . . 6 |- ( ( I e. ( 0..^ M ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) -> ( P ` I ) < ( P ` ( I + 1 ) ) )
76expcom 451 . . . . 5 |- ( A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) -> ( I e. ( 0..^ M ) -> ( P ` I ) < ( P ` ( I + 1 ) ) ) )
87adantl 482 . . . 4 |- ( ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) -> ( I e. ( 0..^ M ) -> ( P ` I ) < ( P ` ( I + 1 ) ) ) )
9 simpl 473 . . . 4 |- ( ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) -> P e. ( RR* ^m ( 0 ... M ) ) )
108, 9jctild 566 . . 3 |- ( ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) -> ( I e. ( 0..^ M ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` I ) < ( P ` ( I + 1 ) ) ) ) )
111, 10syl6bi 243 . 2 |- ( M e. NN -> ( P e. (RePart ` M ) -> ( I e. ( 0..^ M ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` I ) < ( P ` ( I + 1 ) ) ) ) ) )
12113imp 1256 1 |- ( ( M e. NN /\ P e. (RePart ` M ) /\ I e. ( 0..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` I ) < ( P ` ( I + 1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   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:  iccpartgtprec  41356  iccpartipre  41357  iccpartiltu  41358  iccpartigtl  41359  iccpartlt  41360  iccpartgt  41363
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