Theorem pw2dvdslemn 12695

Description: Lemma for pw2dvds 12696. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)

Assertion

Ref	Expression
pw2dvdslemn	|- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )

Distinct variable group:   m, N

Allowed substitution hint:   A( m)

Proof of Theorem pw2dvdslemn

Dummy variables w k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpb 1019	. 2 |- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) -> ( N e. NN /\ -. ( 2 ^ A ) || N ) )
2		oveq2 6015	. . . . . . . 8 |- ( w = 1 -> ( 2 ^ w ) = ( 2 ^ 1 ) )
3	2	breq1d 4093	. . . . . . 7 |- ( w = 1 -> ( ( 2 ^ w ) || N <-> ( 2 ^ 1 ) || N ) )
4	3	notbid 671	. . . . . 6 |- ( w = 1 -> ( -. ( 2 ^ w ) || N <-> -. ( 2 ^ 1 ) || N ) )
5	4	anbi2d 464	. . . . 5 |- ( w = 1 -> ( ( N e. NN /\ -. ( 2 ^ w ) || N ) <-> ( N e. NN /\ -. ( 2 ^ 1 ) || N ) ) )
6	5	imbi1d 231	. . . 4 |- ( w = 1 -> ( ( ( N e. NN /\ -. ( 2 ^ w ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) <-> ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) )
7		oveq2 6015	. . . . . . . 8 |- ( w = k -> ( 2 ^ w ) = ( 2 ^ k ) )
8	7	breq1d 4093	. . . . . . 7 |- ( w = k -> ( ( 2 ^ w ) || N <-> ( 2 ^ k ) || N ) )
9	8	notbid 671	. . . . . 6 |- ( w = k -> ( -. ( 2 ^ w ) || N <-> -. ( 2 ^ k ) || N ) )
10	9	anbi2d 464	. . . . 5 |- ( w = k -> ( ( N e. NN /\ -. ( 2 ^ w ) || N ) <-> ( N e. NN /\ -. ( 2 ^ k ) || N ) ) )
11	10	imbi1d 231	. . . 4 |- ( w = k -> ( ( ( N e. NN /\ -. ( 2 ^ w ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) <-> ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) )
12		oveq2 6015	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( 2 ^ w ) = ( 2 ^ ( k + 1 ) ) )
13	12	breq1d 4093	. . . . . . 7 |- ( w = ( k + 1 ) -> ( ( 2 ^ w ) || N <-> ( 2 ^ ( k + 1 ) ) || N ) )
14	13	notbid 671	. . . . . 6 |- ( w = ( k + 1 ) -> ( -. ( 2 ^ w ) || N <-> -. ( 2 ^ ( k + 1 ) ) || N ) )
15	14	anbi2d 464	. . . . 5 |- ( w = ( k + 1 ) -> ( ( N e. NN /\ -. ( 2 ^ w ) || N ) <-> ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) )
16	15	imbi1d 231	. . . 4 |- ( w = ( k + 1 ) -> ( ( ( N e. NN /\ -. ( 2 ^ w ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) <-> ( ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) )
17		oveq2 6015	. . . . . . . 8 |- ( w = A -> ( 2 ^ w ) = ( 2 ^ A ) )
18	17	breq1d 4093	. . . . . . 7 |- ( w = A -> ( ( 2 ^ w ) || N <-> ( 2 ^ A ) || N ) )
19	18	notbid 671	. . . . . 6 |- ( w = A -> ( -. ( 2 ^ w ) || N <-> -. ( 2 ^ A ) || N ) )
20	19	anbi2d 464	. . . . 5 |- ( w = A -> ( ( N e. NN /\ -. ( 2 ^ w ) || N ) <-> ( N e. NN /\ -. ( 2 ^ A ) || N ) ) )
21	20	imbi1d 231	. . . 4 |- ( w = A -> ( ( ( N e. NN /\ -. ( 2 ^ w ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) <-> ( ( N e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) )
22		0nn0 9392	. . . . . 6 |- 0 e. NN0
23	22	a1i 9	. . . . 5 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> 0 e. NN0 )
24		oveq2 6015	. . . . . . . 8 |- ( m = 0 -> ( 2 ^ m ) = ( 2 ^ 0 ) )
25	24	breq1d 4093	. . . . . . 7 |- ( m = 0 -> ( ( 2 ^ m ) || N <-> ( 2 ^ 0 ) || N ) )
26		oveq1 6014	. . . . . . . . . 10 |- ( m = 0 -> ( m + 1 ) = ( 0 + 1 ) )
27	26	oveq2d 6023	. . . . . . . . 9 |- ( m = 0 -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( 0 + 1 ) ) )
28	27	breq1d 4093	. . . . . . . 8 |- ( m = 0 -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( 0 + 1 ) ) || N ) )
29	28	notbid 671	. . . . . . 7 |- ( m = 0 -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( 0 + 1 ) ) || N ) )
30	25, 29	anbi12d 473	. . . . . 6 |- ( m = 0 -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ 0 ) || N /\ -. ( 2 ^ ( 0 + 1 ) ) || N ) ) )
31	30	adantl 277	. . . . 5 |- ( ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) /\ m = 0 ) -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ 0 ) || N /\ -. ( 2 ^ ( 0 + 1 ) ) || N ) ) )
32		2cnd 9191	. . . . . . . 8 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> 2 e. CC )
33	32	exp0d 10897	. . . . . . 7 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> ( 2 ^ 0 ) = 1 )
34		simpl 109	. . . . . . . . 9 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> N e. NN )
35	34	nnzd 9576	. . . . . . . 8 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> N e. ZZ )
36		1dvds 12324	. . . . . . . 8 |- ( N e. ZZ -> 1 || N )
37	35, 36	syl 14	. . . . . . 7 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> 1 || N )
38	33, 37	eqbrtrd 4105	. . . . . 6 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> ( 2 ^ 0 ) || N )
39		simpr 110	. . . . . . 7 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> -. ( 2 ^ 1 ) || N )
40		0p1e1 9232	. . . . . . . . 9 |- ( 0 + 1 ) = 1
41	40	oveq2i 6018	. . . . . . . 8 |- ( 2 ^ ( 0 + 1 ) ) = ( 2 ^ 1 )
42	41	breq1i 4090	. . . . . . 7 |- ( ( 2 ^ ( 0 + 1 ) ) || N <-> ( 2 ^ 1 ) || N )
43	39, 42	sylnibr 681	. . . . . 6 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> -. ( 2 ^ ( 0 + 1 ) ) || N )
44	38, 43	jca 306	. . . . 5 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> ( ( 2 ^ 0 ) || N /\ -. ( 2 ^ ( 0 + 1 ) ) || N ) )
45	23, 31, 44	rspcedvd 2913	. . . 4 |- ( ( N e. NN /\ -. ( 2 ^ 1 ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
46		simpll 527	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> k e. NN )
47	46	nnnn0d 9430	. . . . . . . 8 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> k e. NN0 )
48		oveq2 6015	. . . . . . . . . . 11 |- ( m = k -> ( 2 ^ m ) = ( 2 ^ k ) )
49	48	breq1d 4093	. . . . . . . . . 10 |- ( m = k -> ( ( 2 ^ m ) || N <-> ( 2 ^ k ) || N ) )
50		oveq1 6014	. . . . . . . . . . . . 13 |- ( m = k -> ( m + 1 ) = ( k + 1 ) )
51	50	oveq2d 6023	. . . . . . . . . . . 12 |- ( m = k -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( k + 1 ) ) )
52	51	breq1d 4093	. . . . . . . . . . 11 |- ( m = k -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( k + 1 ) ) || N ) )
53	52	notbid 671	. . . . . . . . . 10 |- ( m = k -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( k + 1 ) ) || N ) )
54	49, 53	anbi12d 473	. . . . . . . . 9 |- ( m = k -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ k ) || N /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) )
55	54	adantl 277	. . . . . . . 8 |- ( ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) /\ m = k ) -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ k ) || N /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) )
56		simpr 110	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> ( 2 ^ k ) || N )
57		simplrr 536	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> -. ( 2 ^ ( k + 1 ) ) || N )
58	56, 57	jca 306	. . . . . . . 8 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> ( ( 2 ^ k ) || N /\ -. ( 2 ^ ( k + 1 ) ) || N ) )
59	47, 55, 58	rspcedvd 2913	. . . . . . 7 |- ( ( ( k e. NN /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
60	59	adantllr 481	. . . . . 6 |- ( ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
61		simprl 529	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> N e. NN )
62	61	anim1i 340	. . . . . . 7 |- ( ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ -. ( 2 ^ k ) || N ) -> ( N e. NN /\ -. ( 2 ^ k ) || N ) )
63		simpllr 534	. . . . . . 7 |- ( ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ -. ( 2 ^ k ) || N ) -> ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) )
64	62, 63	mpd 13	. . . . . 6 |- ( ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
65		2nn 9280	. . . . . . . . 9 |- 2 e. NN
66		simpll 527	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> k e. NN )
67	66	nnnn0d 9430	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> k e. NN0 )
68		nnexpcl 10782	. . . . . . . . 9 |- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN )
69	65, 67, 68	sylancr 414	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> ( 2 ^ k ) e. NN )
70	61	nnzd 9576	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> N e. ZZ )
71		dvdsdc 12317	. . . . . . . 8 |- ( ( ( 2 ^ k ) e. NN /\ N e. ZZ ) -> DECID ( 2 ^ k ) || N )
72	69, 70, 71	syl2anc 411	. . . . . . 7 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> DECID ( 2 ^ k ) || N )
73		exmiddc 841	. . . . . . 7 |- (DECID ( 2 ^ k ) || N -> ( ( 2 ^ k ) || N \/ -. ( 2 ^ k ) || N ) )
74	72, 73	syl 14	. . . . . 6 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> ( ( 2 ^ k ) || N \/ -. ( 2 ^ k ) || N ) )
75	60, 64, 74	mpjaodan 803	. . . . 5 |- ( ( ( k e. NN /\ ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) /\ ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
76	75	exp31 364	. . . 4 |- ( k e. NN -> ( ( ( N e. NN /\ -. ( 2 ^ k ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) -> ( ( N e. NN /\ -. ( 2 ^ ( k + 1 ) ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) ) )
77	6, 11, 16, 21, 45, 76	nnind 9134	. . 3 |- ( A e. NN -> ( ( N e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) )
78	77	3ad2ant2 1043	. 2 |- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) -> ( ( N e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) )
79	1, 78	mpd 13	1 |- ( ( N e. NN /\ A e. NN /\ -. ( 2 ^ A ) || N ) -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4083  (class class class)co 6007   0cc0 8007   1c1 8008   + caddc 8010   NNcn 9118   2c2 9169   NN0cn0 9377   ZZcz 9454   ^cexp 10768   || cdvds 12306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-dvds 12307

This theorem is referenced by:  pw2dvds  12696