Theorem iccpartlt 41360

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 40335 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)

Hypotheses

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

Assertion

Ref	Expression
iccpartlt	|- ( ph -> ( P ` 0 ) < ( P ` M ) )

Proof of Theorem iccpartlt

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.m	. . . . . . 7 |- ( ph -> M e. NN )
2		iccpartgtprec.p	. . . . . . 7 |- ( ph -> P e. (RePart ` M ) )
3		lbfzo0 12507	. . . . . . . 8 |- ( 0 e. ( 0..^ M ) <-> M e. NN )
4	1, 3	sylibr 224	. . . . . . 7 |- ( ph -> 0 e. ( 0..^ M ) )
5		iccpartimp 41353	. . . . . . 7 |- ( ( M e. NN /\ P e. (RePart ` M ) /\ 0 e. ( 0..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` 0 ) < ( P ` ( 0 + 1 ) ) ) )
6	1, 2, 4, 5	syl3anc 1326	. . . . . 6 |- ( ph -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` 0 ) < ( P ` ( 0 + 1 ) ) ) )
7	6	simprd 479	. . . . 5 |- ( ph -> ( P ` 0 ) < ( P ` ( 0 + 1 ) ) )
8	7	adantl 482	. . . 4 |- ( ( M = 1 /\ ph ) -> ( P ` 0 ) < ( P ` ( 0 + 1 ) ) )
9		fveq2 6191	. . . . . 6 |- ( M = 1 -> ( P ` M ) = ( P ` 1 ) )
10		1e0p1 11552	. . . . . . 7 |- 1 = ( 0 + 1 )
11	10	fveq2i 6194	. . . . . 6 |- ( P ` 1 ) = ( P ` ( 0 + 1 ) )
12	9, 11	syl6eq 2672	. . . . 5 |- ( M = 1 -> ( P ` M ) = ( P ` ( 0 + 1 ) ) )
13	12	adantr 481	. . . 4 |- ( ( M = 1 /\ ph ) -> ( P ` M ) = ( P ` ( 0 + 1 ) ) )
14	8, 13	breqtrrd 4681	. . 3 |- ( ( M = 1 /\ ph ) -> ( P ` 0 ) < ( P ` M ) )
15	14	ex 450	. 2 |- ( M = 1 -> ( ph -> ( P ` 0 ) < ( P ` M ) ) )
16	1, 2	iccpartiltu 41358	. . . 4 |- ( ph -> A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) )
17	1, 2	iccpartigtl 41359	. . . 4 |- ( ph -> A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) )
18		1nn 11031	. . . . . . . . . 10 |- 1 e. NN
19	18	a1i 11	. . . . . . . . 9 |- ( ( ph /\ -. M = 1 ) -> 1 e. NN )
20	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. M = 1 ) -> M e. NN )
21		df-ne 2795	. . . . . . . . . . 11 |- ( M =/= 1 <-> -. M = 1 )
22	1	nnge1d 11063	. . . . . . . . . . . 12 |- ( ph -> 1 <_ M )
23		1red 10055	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. RR )
24	1	nnred 11035	. . . . . . . . . . . . . 14 |- ( ph -> M e. RR )
25	23, 24	ltlend 10182	. . . . . . . . . . . . 13 |- ( ph -> ( 1 < M <-> ( 1 <_ M /\ M =/= 1 ) ) )
26	25	biimprd 238	. . . . . . . . . . . 12 |- ( ph -> ( ( 1 <_ M /\ M =/= 1 ) -> 1 < M ) )
27	22, 26	mpand 711	. . . . . . . . . . 11 |- ( ph -> ( M =/= 1 -> 1 < M ) )
28	21, 27	syl5bir 233	. . . . . . . . . 10 |- ( ph -> ( -. M = 1 -> 1 < M ) )
29	28	imp 445	. . . . . . . . 9 |- ( ( ph /\ -. M = 1 ) -> 1 < M )
30		elfzo1 12517	. . . . . . . . 9 |- ( 1 e. ( 1..^ M ) <-> ( 1 e. NN /\ M e. NN /\ 1 < M ) )
31	19, 20, 29, 30	syl3anbrc 1246	. . . . . . . 8 |- ( ( ph /\ -. M = 1 ) -> 1 e. ( 1..^ M ) )
32		fveq2 6191	. . . . . . . . . 10 |- ( i = 1 -> ( P ` i ) = ( P ` 1 ) )
33	32	breq2d 4665	. . . . . . . . 9 |- ( i = 1 -> ( ( P ` 0 ) < ( P ` i ) <-> ( P ` 0 ) < ( P ` 1 ) ) )
34	33	rspcv 3305	. . . . . . . 8 |- ( 1 e. ( 1..^ M ) -> ( A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) -> ( P ` 0 ) < ( P ` 1 ) ) )
35	31, 34	syl 17	. . . . . . 7 |- ( ( ph /\ -. M = 1 ) -> ( A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) -> ( P ` 0 ) < ( P ` 1 ) ) )
36	32	breq1d 4663	. . . . . . . . . . 11 |- ( i = 1 -> ( ( P ` i ) < ( P ` M ) <-> ( P ` 1 ) < ( P ` M ) ) )
37	36	rspcv 3305	. . . . . . . . . 10 |- ( 1 e. ( 1..^ M ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( P ` 1 ) < ( P ` M ) ) )
38	31, 37	syl 17	. . . . . . . . 9 |- ( ( ph /\ -. M = 1 ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( P ` 1 ) < ( P ` M ) ) )
39		nnnn0 11299	. . . . . . . . . . . . . 14 |- ( M e. NN -> M e. NN0 )
40		0elfz 12436	. . . . . . . . . . . . . 14 |- ( M e. NN0 -> 0 e. ( 0 ... M ) )
41	1, 39, 40	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> 0 e. ( 0 ... M ) )
42	1, 2, 41	iccpartxr 41355	. . . . . . . . . . . 12 |- ( ph -> ( P ` 0 ) e. RR* )
43	42	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ -. M = 1 ) -> ( P ` 0 ) e. RR* )
44	2	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ -. M = 1 ) -> P e. (RePart ` M ) )
45		1nn0 11308	. . . . . . . . . . . . . 14 |- 1 e. NN0
46	45	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ -. M = 1 ) -> 1 e. NN0 )
47	1, 39	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> M e. NN0 )
48	47	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ -. M = 1 ) -> M e. NN0 )
49	22	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ -. M = 1 ) -> 1 <_ M )
50		elfz2nn0 12431	. . . . . . . . . . . . 13 |- ( 1 e. ( 0 ... M ) <-> ( 1 e. NN0 /\ M e. NN0 /\ 1 <_ M ) )
51	46, 48, 49, 50	syl3anbrc 1246	. . . . . . . . . . . 12 |- ( ( ph /\ -. M = 1 ) -> 1 e. ( 0 ... M ) )
52	20, 44, 51	iccpartxr 41355	. . . . . . . . . . 11 |- ( ( ph /\ -. M = 1 ) -> ( P ` 1 ) e. RR* )
53		nn0fz0 12437	. . . . . . . . . . . . . . 15 |- ( M e. NN0 <-> M e. ( 0 ... M ) )
54	39, 53	sylib 208	. . . . . . . . . . . . . 14 |- ( M e. NN -> M e. ( 0 ... M ) )
55	1, 54	syl 17	. . . . . . . . . . . . 13 |- ( ph -> M e. ( 0 ... M ) )
56	1, 2, 55	iccpartxr 41355	. . . . . . . . . . . 12 |- ( ph -> ( P ` M ) e. RR* )
57	56	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ -. M = 1 ) -> ( P ` M ) e. RR* )
58		xrlttr 11973	. . . . . . . . . . 11 |- ( ( ( P ` 0 ) e. RR* /\ ( P ` 1 ) e. RR* /\ ( P ` M ) e. RR* ) -> ( ( ( P ` 0 ) < ( P ` 1 ) /\ ( P ` 1 ) < ( P ` M ) ) -> ( P ` 0 ) < ( P ` M ) ) )
59	43, 52, 57, 58	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ -. M = 1 ) -> ( ( ( P ` 0 ) < ( P ` 1 ) /\ ( P ` 1 ) < ( P ` M ) ) -> ( P ` 0 ) < ( P ` M ) ) )
60	59	expcomd 454	. . . . . . . . 9 |- ( ( ph /\ -. M = 1 ) -> ( ( P ` 1 ) < ( P ` M ) -> ( ( P ` 0 ) < ( P ` 1 ) -> ( P ` 0 ) < ( P ` M ) ) ) )
61	38, 60	syld 47	. . . . . . . 8 |- ( ( ph /\ -. M = 1 ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( ( P ` 0 ) < ( P ` 1 ) -> ( P ` 0 ) < ( P ` M ) ) ) )
62	61	com23 86	. . . . . . 7 |- ( ( ph /\ -. M = 1 ) -> ( ( P ` 0 ) < ( P ` 1 ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( P ` 0 ) < ( P ` M ) ) ) )
63	35, 62	syld 47	. . . . . 6 |- ( ( ph /\ -. M = 1 ) -> ( A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( P ` 0 ) < ( P ` M ) ) ) )
64	63	ex 450	. . . . 5 |- ( ph -> ( -. M = 1 -> ( A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( P ` 0 ) < ( P ` M ) ) ) ) )
65	64	com24 95	. . . 4 |- ( ph -> ( A. i e. ( 1..^ M ) ( P ` i ) < ( P ` M ) -> ( A. i e. ( 1..^ M ) ( P ` 0 ) < ( P ` i ) -> ( -. M = 1 -> ( P ` 0 ) < ( P ` M ) ) ) ) )
66	16, 17, 65	mp2d 49	. . 3 |- ( ph -> ( -. M = 1 -> ( P ` 0 ) < ( P ` M ) ) )
67	66	com12 32	. 2 |- ( -. M = 1 -> ( ph -> ( P ` 0 ) < ( P ` M ) ) )
68	15, 67	pm2.61i 176	1 |- ( ph -> ( P ` 0 ) < ( P ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   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   <_ cle 10075   NNcn 11020   NN0cn0 11292   ...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-fz 12327  df-fzo 12466  df-iccp 41350

This theorem is referenced by:  iccpartltu  41361  iccpartgtl  41362  iccpartgt  41363