Theorem iccpartxr 41355

Description: If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)

Hypotheses

Ref	Expression
iccpartgtprec.m	|- ( ph -> M e. NN )
iccpartgtprec.p	|- ( ph -> P e. (RePart ` M ) )
iccpartxr.i	|- ( ph -> I e. ( 0 ... M ) )

Assertion

Ref	Expression
iccpartxr	|- ( ph -> ( P ` I ) e. RR* )

Proof of Theorem iccpartxr

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.p	. . . . 5 |- ( ph -> P e. (RePart ` M ) )
2		iccpartgtprec.m	. . . . . 6 |- ( ph -> M e. NN )
3		iccpart 41352	. . . . . 6 |- ( M e. NN -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )
4	2, 3	syl 17	. . . . 5 |- ( ph -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )
5	1, 4	mpbid 222	. . . 4 |- ( ph -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) )
6	5	simpld 475	. . 3 |- ( ph -> P e. ( RR* ^m ( 0 ... M ) ) )
7		elmapi 7879	. . 3 |- ( P e. ( RR* ^m ( 0 ... M ) ) -> P : ( 0 ... M ) --> RR* )
8	6, 7	syl 17	. 2 |- ( ph -> P : ( 0 ... M ) --> RR* )
9		iccpartxr.i	. 2 |- ( ph -> I e. ( 0 ... M ) )
10	8, 9	ffvelrnd 6360	1 |- ( ph -> ( P ` I ) e. RR* )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   class class class wbr 4653   -->wf 5884   ` 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-iccp 41350

This theorem is referenced by:  iccpartipre  41357  iccpartiltu  41358  iccpartigtl  41359  iccpartlt  41360  iccpartleu  41364  iccpartgel  41365  iccpartrn  41366  iccelpart  41369  iccpartiun  41370  icceuelpartlem  41371  icceuelpart  41372  iccpartdisj  41373  iccpartnel  41374  bgoldbtbndlem2  41694