Theorem irrapxlem1 37386

Description: Lemma for irrapx1 37392. Divides the unit interval into B half-open sections and using the pigeonhole principle fphpdo 37381 finds two multiples of A in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Assertion

Ref	Expression
irrapxlem1	|- ( ( A e. RR+ /\ B e. NN ) -> E. x e. ( 0 ... B ) E. y e. ( 0 ... B ) ( x < y /\ ( |_ ` ( B x. ( ( A x. x ) mod 1 ) ) ) = ( |_ ` ( B x. ( ( A x. y ) mod 1 ) ) ) ) )

Distinct variable groups:   x, A, y   x, B, y

Proof of Theorem irrapxlem1

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzssuz 12382	. . . 4 |- ( 0 ... B ) C_ ( ZZ>= ` 0 )
2		uzssz 11707	. . . . 5 |- ( ZZ>= ` 0 ) C_ ZZ
3		zssre 11384	. . . . 5 |- ZZ C_ RR
4	2, 3	sstri 3612	. . . 4 |- ( ZZ>= ` 0 ) C_ RR
5	1, 4	sstri 3612	. . 3 |- ( 0 ... B ) C_ RR
6	5	a1i 11	. 2 |- ( ( A e. RR+ /\ B e. NN ) -> ( 0 ... B ) C_ RR )
7		ovexd 6680	. 2 |- ( ( A e. RR+ /\ B e. NN ) -> ( 0 ... ( B - 1 ) ) e. _V )
8		nnm1nn0 11334	. . . . 5 |- ( B e. NN -> ( B - 1 ) e. NN0 )
9	8	adantl 482	. . . 4 |- ( ( A e. RR+ /\ B e. NN ) -> ( B - 1 ) e. NN0 )
10		nn0uz 11722	. . . 4 |- NN0 = ( ZZ>= ` 0 )
11	9, 10	syl6eleq 2711	. . 3 |- ( ( A e. RR+ /\ B e. NN ) -> ( B - 1 ) e. ( ZZ>= ` 0 ) )
12		nnz 11399	. . . 4 |- ( B e. NN -> B e. ZZ )
13	12	adantl 482	. . 3 |- ( ( A e. RR+ /\ B e. NN ) -> B e. ZZ )
14		nnre 11027	. . . . 5 |- ( B e. NN -> B e. RR )
15	14	adantl 482	. . . 4 |- ( ( A e. RR+ /\ B e. NN ) -> B e. RR )
16	15	ltm1d 10956	. . 3 |- ( ( A e. RR+ /\ B e. NN ) -> ( B - 1 ) < B )
17		fzsdom2 13215	. . 3 |- ( ( ( ( B - 1 ) e. ( ZZ>= ` 0 ) /\ B e. ZZ ) /\ ( B - 1 ) < B ) -> ( 0 ... ( B - 1 ) ) ~< ( 0 ... B ) )
18	11, 13, 16, 17	syl21anc 1325	. 2 |- ( ( A e. RR+ /\ B e. NN ) -> ( 0 ... ( B - 1 ) ) ~< ( 0 ... B ) )
19	14	ad2antlr 763	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> B e. RR )
20		rpre 11839	. . . . . . . . 9 |- ( A e. RR+ -> A e. RR )
21	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> A e. RR )
22		elfzelz 12342	. . . . . . . . . 10 |- ( a e. ( 0 ... B ) -> a e. ZZ )
23	22	zred 11482	. . . . . . . . 9 |- ( a e. ( 0 ... B ) -> a e. RR )
24	23	adantl 482	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> a e. RR )
25	21, 24	remulcld 10070	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( A x. a ) e. RR )
26		1rp 11836	. . . . . . 7 |- 1 e. RR+
27		modcl 12672	. . . . . . 7 |- ( ( ( A x. a ) e. RR /\ 1 e. RR+ ) -> ( ( A x. a ) mod 1 ) e. RR )
28	25, 26, 27	sylancl 694	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( A x. a ) mod 1 ) e. RR )
29	19, 28	remulcld 10070	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. ( ( A x. a ) mod 1 ) ) e. RR )
30	29	flcld 12599	. . . 4 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. ZZ )
31	19	recnd 10068	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> B e. CC )
32	31	mul01d 10235	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. 0 ) = 0 )
33		modge0 12678	. . . . . . . . . 10 |- ( ( ( A x. a ) e. RR /\ 1 e. RR+ ) -> 0 <_ ( ( A x. a ) mod 1 ) )
34	25, 26, 33	sylancl 694	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 0 <_ ( ( A x. a ) mod 1 ) )
35		0red 10041	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 0 e. RR )
36		nngt0 11049	. . . . . . . . . . 11 |- ( B e. NN -> 0 < B )
37	36	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 0 < B )
38		lemul2 10876	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( ( A x. a ) mod 1 ) e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( 0 <_ ( ( A x. a ) mod 1 ) <-> ( B x. 0 ) <_ ( B x. ( ( A x. a ) mod 1 ) ) ) )
39	35, 28, 19, 37, 38	syl112anc 1330	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( 0 <_ ( ( A x. a ) mod 1 ) <-> ( B x. 0 ) <_ ( B x. ( ( A x. a ) mod 1 ) ) ) )
40	34, 39	mpbid 222	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. 0 ) <_ ( B x. ( ( A x. a ) mod 1 ) ) )
41	32, 40	eqbrtrrd 4677	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 0 <_ ( B x. ( ( A x. a ) mod 1 ) ) )
42	35, 29	lenltd 10183	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( 0 <_ ( B x. ( ( A x. a ) mod 1 ) ) <-> -. ( B x. ( ( A x. a ) mod 1 ) ) < 0 ) )
43	41, 42	mpbid 222	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> -. ( B x. ( ( A x. a ) mod 1 ) ) < 0 )
44		0z 11388	. . . . . . 7 |- 0 e. ZZ
45		fllt 12607	. . . . . . 7 |- ( ( ( B x. ( ( A x. a ) mod 1 ) ) e. RR /\ 0 e. ZZ ) -> ( ( B x. ( ( A x. a ) mod 1 ) ) < 0 <-> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < 0 ) )
46	29, 44, 45	sylancl 694	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( B x. ( ( A x. a ) mod 1 ) ) < 0 <-> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < 0 ) )
47	43, 46	mtbid 314	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> -. ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < 0 )
48	30	zred 11482	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. RR )
49	35, 48	lenltd 10183	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( 0 <_ ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <-> -. ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < 0 ) )
50	47, 49	mpbird 247	. . . 4 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 0 <_ ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) )
51		elnn0z 11390	. . . 4 |- ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. NN0 <-> ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. ZZ /\ 0 <_ ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) ) )
52	30, 50, 51	sylanbrc 698	. . 3 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. NN0 )
53	8	ad2antlr 763	. . 3 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B - 1 ) e. NN0 )
54		flle 12600	. . . . . . 7 |- ( ( B x. ( ( A x. a ) mod 1 ) ) e. RR -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B x. ( ( A x. a ) mod 1 ) ) )
55	29, 54	syl 17	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B x. ( ( A x. a ) mod 1 ) ) )
56		modlt 12679	. . . . . . . . 9 |- ( ( ( A x. a ) e. RR /\ 1 e. RR+ ) -> ( ( A x. a ) mod 1 ) < 1 )
57	25, 26, 56	sylancl 694	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( A x. a ) mod 1 ) < 1 )
58		1red 10055	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> 1 e. RR )
59		ltmul2 10874	. . . . . . . . 9 |- ( ( ( ( A x. a ) mod 1 ) e. RR /\ 1 e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( ( A x. a ) mod 1 ) < 1 <-> ( B x. ( ( A x. a ) mod 1 ) ) < ( B x. 1 ) ) )
60	28, 58, 19, 37, 59	syl112anc 1330	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( ( A x. a ) mod 1 ) < 1 <-> ( B x. ( ( A x. a ) mod 1 ) ) < ( B x. 1 ) ) )
61	57, 60	mpbid 222	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. ( ( A x. a ) mod 1 ) ) < ( B x. 1 ) )
62	31	mulid1d 10057	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. 1 ) = B )
63	61, 62	breqtrd 4679	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B x. ( ( A x. a ) mod 1 ) ) < B )
64	48, 29, 19, 55, 63	lelttrd 10195	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < B )
65		nncn 11028	. . . . . . 7 |- ( B e. NN -> B e. CC )
66		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
67		npcan 10290	. . . . . . 7 |- ( ( B e. CC /\ 1 e. CC ) -> ( ( B - 1 ) + 1 ) = B )
68	65, 66, 67	sylancl 694	. . . . . 6 |- ( B e. NN -> ( ( B - 1 ) + 1 ) = B )
69	68	ad2antlr 763	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( B - 1 ) + 1 ) = B )
70	64, 69	breqtrrd 4681	. . . 4 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < ( ( B - 1 ) + 1 ) )
71	12	ad2antlr 763	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> B e. ZZ )
72		1z 11407	. . . . . 6 |- 1 e. ZZ
73		zsubcl 11419	. . . . . 6 |- ( ( B e. ZZ /\ 1 e. ZZ ) -> ( B - 1 ) e. ZZ )
74	71, 72, 73	sylancl 694	. . . . 5 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( B - 1 ) e. ZZ )
75		zleltp1 11428	. . . . 5 |- ( ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. ZZ /\ ( B - 1 ) e. ZZ ) -> ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B - 1 ) <-> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < ( ( B - 1 ) + 1 ) ) )
76	30, 74, 75	syl2anc 693	. . . 4 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B - 1 ) <-> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) < ( ( B - 1 ) + 1 ) ) )
77	70, 76	mpbird 247	. . 3 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B - 1 ) )
78		elfz2nn0 12431	. . 3 |- ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. ( 0 ... ( B - 1 ) ) <-> ( ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. NN0 /\ ( B - 1 ) e. NN0 /\ ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) <_ ( B - 1 ) ) )
79	52, 53, 77, 78	syl3anbrc 1246	. 2 |- ( ( ( A e. RR+ /\ B e. NN ) /\ a e. ( 0 ... B ) ) -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) e. ( 0 ... ( B - 1 ) ) )
80		oveq2 6658	. . . . 5 |- ( a = x -> ( A x. a ) = ( A x. x ) )
81	80	oveq1d 6665	. . . 4 |- ( a = x -> ( ( A x. a ) mod 1 ) = ( ( A x. x ) mod 1 ) )
82	81	oveq2d 6666	. . 3 |- ( a = x -> ( B x. ( ( A x. a ) mod 1 ) ) = ( B x. ( ( A x. x ) mod 1 ) ) )
83	82	fveq2d 6195	. 2 |- ( a = x -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) = ( |_ ` ( B x. ( ( A x. x ) mod 1 ) ) ) )
84		oveq2 6658	. . . . 5 |- ( a = y -> ( A x. a ) = ( A x. y ) )
85	84	oveq1d 6665	. . . 4 |- ( a = y -> ( ( A x. a ) mod 1 ) = ( ( A x. y ) mod 1 ) )
86	85	oveq2d 6666	. . 3 |- ( a = y -> ( B x. ( ( A x. a ) mod 1 ) ) = ( B x. ( ( A x. y ) mod 1 ) ) )
87	86	fveq2d 6195	. 2 |- ( a = y -> ( |_ ` ( B x. ( ( A x. a ) mod 1 ) ) ) = ( |_ ` ( B x. ( ( A x. y ) mod 1 ) ) ) )
88	6, 7, 18, 79, 83, 87	fphpdo 37381	1 |- ( ( A e. RR+ /\ B e. NN ) -> E. x e. ( 0 ... B ) E. y e. ( 0 ... B ) ( x < y /\ ( |_ ` ( B x. ( ( A x. x ) mod 1 ) ) ) = ( |_ ` ( B x. ( ( A x. y ) mod 1 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~< csdm 7954   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   |_cfl 12591   mod cmo 12668

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-rep 4771  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  ax-pre-sup 10014

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-rmo 2920  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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-hash 13118

This theorem is referenced by:  irrapxlem2  37387