Theorem icceuelpartlem 41371

Description: Lemma for icceuelpart 41372. (Contributed by AV, 19-Jul-2020.)

Hypotheses

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

Assertion

Ref	Expression
icceuelpartlem	|- ( ph -> ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> ( I < J -> ( P ` ( I + 1 ) ) <_ ( P ` J ) ) ) )

Proof of Theorem icceuelpartlem

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( ( I + 1 ) = J -> ( P ` ( I + 1 ) ) = ( P ` J ) )
2	1	olcd 408	. . . . 5 |- ( ( I + 1 ) = J -> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) )
3	2	a1d 25	. . . 4 |- ( ( I + 1 ) = J -> ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) ) )
4		elfzoelz 12470	. . . . . . . . . . 11 |- ( I e. ( 0..^ M ) -> I e. ZZ )
5		elfzoelz 12470	. . . . . . . . . . 11 |- ( J e. ( 0..^ M ) -> J e. ZZ )
6		zltp1le 11427	. . . . . . . . . . . . . . . . 17 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I < J <-> ( I + 1 ) <_ J ) )
7	6	biimpcd 239	. . . . . . . . . . . . . . . 16 |- ( I < J -> ( ( I e. ZZ /\ J e. ZZ ) -> ( I + 1 ) <_ J ) )
8	7	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( I < J /\ -. ( I + 1 ) = J ) -> ( ( I e. ZZ /\ J e. ZZ ) -> ( I + 1 ) <_ J ) )
9	8	impcom 446	. . . . . . . . . . . . . 14 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> ( I + 1 ) <_ J )
10		df-ne 2795	. . . . . . . . . . . . . . . . 17 |- ( ( I + 1 ) =/= J <-> -. ( I + 1 ) = J )
11		necom 2847	. . . . . . . . . . . . . . . . 17 |- ( ( I + 1 ) =/= J <-> J =/= ( I + 1 ) )
12	10, 11	sylbb1 227	. . . . . . . . . . . . . . . 16 |- ( -. ( I + 1 ) = J -> J =/= ( I + 1 ) )
13	12	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( I < J /\ -. ( I + 1 ) = J ) -> J =/= ( I + 1 ) )
14	13	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> J =/= ( I + 1 ) )
15	9, 14	jca 554	. . . . . . . . . . . . 13 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> ( ( I + 1 ) <_ J /\ J =/= ( I + 1 ) ) )
16		peano2z 11418	. . . . . . . . . . . . . . . . 17 |- ( I e. ZZ -> ( I + 1 ) e. ZZ )
17	16	zred 11482	. . . . . . . . . . . . . . . 16 |- ( I e. ZZ -> ( I + 1 ) e. RR )
18		zre 11381	. . . . . . . . . . . . . . . 16 |- ( J e. ZZ -> J e. RR )
19	17, 18	anim12i 590	. . . . . . . . . . . . . . 15 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( ( I + 1 ) e. RR /\ J e. RR ) )
20	19	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> ( ( I + 1 ) e. RR /\ J e. RR ) )
21		ltlen 10138	. . . . . . . . . . . . . 14 |- ( ( ( I + 1 ) e. RR /\ J e. RR ) -> ( ( I + 1 ) < J <-> ( ( I + 1 ) <_ J /\ J =/= ( I + 1 ) ) ) )
22	20, 21	syl 17	. . . . . . . . . . . . 13 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> ( ( I + 1 ) < J <-> ( ( I + 1 ) <_ J /\ J =/= ( I + 1 ) ) ) )
23	15, 22	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( I < J /\ -. ( I + 1 ) = J ) ) -> ( I + 1 ) < J )
24	23	ex 450	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( ( I < J /\ -. ( I + 1 ) = J ) -> ( I + 1 ) < J ) )
25	4, 5, 24	syl2an 494	. . . . . . . . . 10 |- ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> ( ( I < J /\ -. ( I + 1 ) = J ) -> ( I + 1 ) < J ) )
26	25	adantl 482	. . . . . . . . 9 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( ( I < J /\ -. ( I + 1 ) = J ) -> ( I + 1 ) < J ) )
27		iccpartiun.m	. . . . . . . . . . 11 |- ( ph -> M e. NN )
28		iccpartiun.p	. . . . . . . . . . 11 |- ( ph -> P e. (RePart ` M ) )
29	27, 28	iccpartgt 41363	. . . . . . . . . 10 |- ( ph -> A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( i < j -> ( P ` i ) < ( P ` j ) ) )
30		fzofzp1 12565	. . . . . . . . . . 11 |- ( I e. ( 0..^ M ) -> ( I + 1 ) e. ( 0 ... M ) )
31		elfzofz 12485	. . . . . . . . . . 11 |- ( J e. ( 0..^ M ) -> J e. ( 0 ... M ) )
32		breq1 4656	. . . . . . . . . . . . 13 |- ( i = ( I + 1 ) -> ( i < j <-> ( I + 1 ) < j ) )
33		fveq2 6191	. . . . . . . . . . . . . 14 |- ( i = ( I + 1 ) -> ( P ` i ) = ( P ` ( I + 1 ) ) )
34	33	breq1d 4663	. . . . . . . . . . . . 13 |- ( i = ( I + 1 ) -> ( ( P ` i ) < ( P ` j ) <-> ( P ` ( I + 1 ) ) < ( P ` j ) ) )
35	32, 34	imbi12d 334	. . . . . . . . . . . 12 |- ( i = ( I + 1 ) -> ( ( i < j -> ( P ` i ) < ( P ` j ) ) <-> ( ( I + 1 ) < j -> ( P ` ( I + 1 ) ) < ( P ` j ) ) ) )
36		breq2 4657	. . . . . . . . . . . . 13 |- ( j = J -> ( ( I + 1 ) < j <-> ( I + 1 ) < J ) )
37		fveq2 6191	. . . . . . . . . . . . . 14 |- ( j = J -> ( P ` j ) = ( P ` J ) )
38	37	breq2d 4665	. . . . . . . . . . . . 13 |- ( j = J -> ( ( P ` ( I + 1 ) ) < ( P ` j ) <-> ( P ` ( I + 1 ) ) < ( P ` J ) ) )
39	36, 38	imbi12d 334	. . . . . . . . . . . 12 |- ( j = J -> ( ( ( I + 1 ) < j -> ( P ` ( I + 1 ) ) < ( P ` j ) ) <-> ( ( I + 1 ) < J -> ( P ` ( I + 1 ) ) < ( P ` J ) ) ) )
40	35, 39	rspc2v 3322	. . . . . . . . . . 11 |- ( ( ( I + 1 ) e. ( 0 ... M ) /\ J e. ( 0 ... M ) ) -> ( A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( i < j -> ( P ` i ) < ( P ` j ) ) -> ( ( I + 1 ) < J -> ( P ` ( I + 1 ) ) < ( P ` J ) ) ) )
41	30, 31, 40	syl2an 494	. . . . . . . . . 10 |- ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> ( A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( i < j -> ( P ` i ) < ( P ` j ) ) -> ( ( I + 1 ) < J -> ( P ` ( I + 1 ) ) < ( P ` J ) ) ) )
42	29, 41	mpan9 486	. . . . . . . . 9 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( ( I + 1 ) < J -> ( P ` ( I + 1 ) ) < ( P ` J ) ) )
43	26, 42	syld 47	. . . . . . . 8 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( ( I < J /\ -. ( I + 1 ) = J ) -> ( P ` ( I + 1 ) ) < ( P ` J ) ) )
44	43	expdimp 453	. . . . . . 7 |- ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( -. ( I + 1 ) = J -> ( P ` ( I + 1 ) ) < ( P ` J ) ) )
45	44	impcom 446	. . . . . 6 |- ( ( -. ( I + 1 ) = J /\ ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) ) -> ( P ` ( I + 1 ) ) < ( P ` J ) )
46	45	orcd 407	. . . . 5 |- ( ( -. ( I + 1 ) = J /\ ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) ) -> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) )
47	46	ex 450	. . . 4 |- ( -. ( I + 1 ) = J -> ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) ) )
48	3, 47	pm2.61i 176	. . 3 |- ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) )
49	27	adantr 481	. . . . . . 7 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> M e. NN )
50	28	adantr 481	. . . . . . 7 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> P e. (RePart ` M ) )
51	30	adantr 481	. . . . . . . 8 |- ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> ( I + 1 ) e. ( 0 ... M ) )
52	51	adantl 482	. . . . . . 7 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( I + 1 ) e. ( 0 ... M ) )
53	49, 50, 52	iccpartxr 41355	. . . . . 6 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( P ` ( I + 1 ) ) e. RR* )
54	31	adantl 482	. . . . . . . 8 |- ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> J e. ( 0 ... M ) )
55	54	adantl 482	. . . . . . 7 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> J e. ( 0 ... M ) )
56	49, 50, 55	iccpartxr 41355	. . . . . 6 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( P ` J ) e. RR* )
57	53, 56	jca 554	. . . . 5 |- ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) -> ( ( P ` ( I + 1 ) ) e. RR* /\ ( P ` J ) e. RR* ) )
58	57	adantr 481	. . . 4 |- ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( ( P ` ( I + 1 ) ) e. RR* /\ ( P ` J ) e. RR* ) )
59		xrleloe 11977	. . . 4 |- ( ( ( P ` ( I + 1 ) ) e. RR* /\ ( P ` J ) e. RR* ) -> ( ( P ` ( I + 1 ) ) <_ ( P ` J ) <-> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) ) )
60	58, 59	syl 17	. . 3 |- ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( ( P ` ( I + 1 ) ) <_ ( P ` J ) <-> ( ( P ` ( I + 1 ) ) < ( P ` J ) \/ ( P ` ( I + 1 ) ) = ( P ` J ) ) ) )
61	48, 60	mpbird 247	. 2 |- ( ( ( ph /\ ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) ) /\ I < J ) -> ( P ` ( I + 1 ) ) <_ ( P ` J ) )
62	61	exp31 630	1 |- ( ph -> ( ( I e. ( 0..^ M ) /\ J e. ( 0..^ M ) ) -> ( I < J -> ( P ` ( I + 1 ) ) <_ ( P ` J ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   ZZcz 11377   ...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:  icceuelpart  41372