Theorem iccpartrn 41366

Description: If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)

Hypotheses

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

Assertion

Ref	Expression
iccpartrn	|- ( ph -> ran P C_ ( ( P ` 0 ) [,] ( P ` M ) ) )

Proof of Theorem iccpartrn

Dummy variables k i p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.p	. . . . 5 |- ( ph -> P e. (RePart ` M ) )
2		iccpartgtprec.m	. . . . . . 7 |- ( ph -> M e. NN )
3		iccpart 41352	. . . . . . 7 |- ( 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	. . . . . 6 |- ( ph -> ( P e. (RePart ` M ) <-> ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) ) )
5		elmapfn 7880	. . . . . . 7 |- ( P e. ( RR* ^m ( 0 ... M ) ) -> P Fn ( 0 ... M ) )
6	5	adantr 481	. . . . . 6 |- ( ( P e. ( RR* ^m ( 0 ... M ) ) /\ A. i e. ( 0..^ M ) ( P ` i ) < ( P ` ( i + 1 ) ) ) -> P Fn ( 0 ... M ) )
7	4, 6	syl6bi 243	. . . . 5 |- ( ph -> ( P e. (RePart ` M ) -> P Fn ( 0 ... M ) ) )
8	1, 7	mpd 15	. . . 4 |- ( ph -> P Fn ( 0 ... M ) )
9		fvelrnb 6243	. . . 4 |- ( P Fn ( 0 ... M ) -> ( p e. ran P <-> E. i e. ( 0 ... M ) ( P ` i ) = p ) )
10	8, 9	syl 17	. . 3 |- ( ph -> ( p e. ran P <-> E. i e. ( 0 ... M ) ( P ` i ) = p ) )
11	2	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) ) -> M e. NN )
12	1	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) ) -> P e. (RePart ` M ) )
13		simpr 477	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) ) -> i e. ( 0 ... M ) )
14	11, 12, 13	iccpartxr 41355	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( P ` i ) e. RR* )
15	2, 1	iccpartgel 41365	. . . . . . . 8 |- ( ph -> A. k e. ( 0 ... M ) ( P ` 0 ) <_ ( P ` k ) )
16		fveq2 6191	. . . . . . . . . . 11 |- ( k = i -> ( P ` k ) = ( P ` i ) )
17	16	breq2d 4665	. . . . . . . . . 10 |- ( k = i -> ( ( P ` 0 ) <_ ( P ` k ) <-> ( P ` 0 ) <_ ( P ` i ) ) )
18	17	rspcva 3307	. . . . . . . . 9 |- ( ( i e. ( 0 ... M ) /\ A. k e. ( 0 ... M ) ( P ` 0 ) <_ ( P ` k ) ) -> ( P ` 0 ) <_ ( P ` i ) )
19	18	expcom 451	. . . . . . . 8 |- ( A. k e. ( 0 ... M ) ( P ` 0 ) <_ ( P ` k ) -> ( i e. ( 0 ... M ) -> ( P ` 0 ) <_ ( P ` i ) ) )
20	15, 19	syl 17	. . . . . . 7 |- ( ph -> ( i e. ( 0 ... M ) -> ( P ` 0 ) <_ ( P ` i ) ) )
21	20	imp 445	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( P ` 0 ) <_ ( P ` i ) )
22	2, 1	iccpartleu 41364	. . . . . . . 8 |- ( ph -> A. k e. ( 0 ... M ) ( P ` k ) <_ ( P ` M ) )
23	16	breq1d 4663	. . . . . . . . . 10 |- ( k = i -> ( ( P ` k ) <_ ( P ` M ) <-> ( P ` i ) <_ ( P ` M ) ) )
24	23	rspcva 3307	. . . . . . . . 9 |- ( ( i e. ( 0 ... M ) /\ A. k e. ( 0 ... M ) ( P ` k ) <_ ( P ` M ) ) -> ( P ` i ) <_ ( P ` M ) )
25	24	expcom 451	. . . . . . . 8 |- ( A. k e. ( 0 ... M ) ( P ` k ) <_ ( P ` M ) -> ( i e. ( 0 ... M ) -> ( P ` i ) <_ ( P ` M ) ) )
26	22, 25	syl 17	. . . . . . 7 |- ( ph -> ( i e. ( 0 ... M ) -> ( P ` i ) <_ ( P ` M ) ) )
27	26	imp 445	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( P ` i ) <_ ( P ` M ) )
28		nnnn0 11299	. . . . . . . . . . 11 |- ( M e. NN -> M e. NN0 )
29		0elfz 12436	. . . . . . . . . . 11 |- ( M e. NN0 -> 0 e. ( 0 ... M ) )
30	2, 28, 29	3syl 18	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 ... M ) )
31	2, 1, 30	iccpartxr 41355	. . . . . . . . 9 |- ( ph -> ( P ` 0 ) e. RR* )
32		nn0fz0 12437	. . . . . . . . . . . 12 |- ( M e. NN0 <-> M e. ( 0 ... M ) )
33	28, 32	sylib 208	. . . . . . . . . . 11 |- ( M e. NN -> M e. ( 0 ... M ) )
34	2, 33	syl 17	. . . . . . . . . 10 |- ( ph -> M e. ( 0 ... M ) )
35	2, 1, 34	iccpartxr 41355	. . . . . . . . 9 |- ( ph -> ( P ` M ) e. RR* )
36	31, 35	jca 554	. . . . . . . 8 |- ( ph -> ( ( P ` 0 ) e. RR* /\ ( P ` M ) e. RR* ) )
37	36	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( P ` 0 ) e. RR* /\ ( P ` M ) e. RR* ) )
38		elicc1 12219	. . . . . . 7 |- ( ( ( P ` 0 ) e. RR* /\ ( P ` M ) e. RR* ) -> ( ( P ` i ) e. ( ( P ` 0 ) [,] ( P ` M ) ) <-> ( ( P ` i ) e. RR* /\ ( P ` 0 ) <_ ( P ` i ) /\ ( P ` i ) <_ ( P ` M ) ) ) )
39	37, 38	syl 17	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( P ` i ) e. ( ( P ` 0 ) [,] ( P ` M ) ) <-> ( ( P ` i ) e. RR* /\ ( P ` 0 ) <_ ( P ` i ) /\ ( P ` i ) <_ ( P ` M ) ) ) )
40	14, 21, 27, 39	mpbir3and 1245	. . . . 5 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( P ` i ) e. ( ( P ` 0 ) [,] ( P ` M ) ) )
41		eleq1 2689	. . . . 5 |- ( ( P ` i ) = p -> ( ( P ` i ) e. ( ( P ` 0 ) [,] ( P ` M ) ) <-> p e. ( ( P ` 0 ) [,] ( P ` M ) ) ) )
42	40, 41	syl5ibcom 235	. . . 4 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( P ` i ) = p -> p e. ( ( P ` 0 ) [,] ( P ` M ) ) ) )
43	42	rexlimdva 3031	. . 3 |- ( ph -> ( E. i e. ( 0 ... M ) ( P ` i ) = p -> p e. ( ( P ` 0 ) [,] ( P ` M ) ) ) )
44	10, 43	sylbid 230	. 2 |- ( ph -> ( p e. ran P -> p e. ( ( P ` 0 ) [,] ( P ` M ) ) ) )
45	44	ssrdv 3609	1 |- ( ph -> ran P C_ ( ( P ` 0 ) [,] ( P ` M ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ran crn 5115   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   [,]cicc 12178   ...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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-icc 12182  df-fz 12327  df-fzo 12466  df-iccp 41350

This theorem is referenced by:  iccpartf  41367