Theorem lgsdilem2 16035

Description: Lemma for lgsdi 16036. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
lgsdilem2.1	|- ( ph -> A e. ZZ )
lgsdilem2.2	|- ( ph -> M e. ZZ )
lgsdilem2.3	|- ( ph -> N e. ZZ )
lgsdilem2.4	|- ( ph -> M =/= 0 )
lgsdilem2.5	|- ( ph -> N =/= 0 )
lgsdilem2.6	|- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )

Assertion

Ref	Expression
lgsdilem2	|- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Distinct variable groups:   n, M   A, n   n, N

Allowed substitution hints:   ph( n)   F( n)

Proof of Theorem lgsdilem2

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

Step	Hyp	Ref	Expression
1		mulrid 8287	. . 3 |- ( k e. CC -> ( k x. 1 ) = k )
2	1	adantl 277	. 2 |- ( ( ph /\ k e. CC ) -> ( k x. 1 ) = k )
3		lgsdilem2.2	. . . 4 |- ( ph -> M e. ZZ )
4		lgsdilem2.4	. . . 4 |- ( ph -> M =/= 0 )
5		nnabscl 11810	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 411	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 9908	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	eleqtrdi 2327	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 9717	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 9724	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 9719	. . . . . . 7 |- ( ph -> M e. CC )
13	10	zcnd 9719	. . . . . . 7 |- ( ph -> N e. CC )
14		0z 9605	. . . . . . . . 9 |- 0 e. ZZ
15		zapne 9669	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
16	3, 14, 15	sylancl 413	. . . . . . . 8 |- ( ph -> ( M # 0 <-> M =/= 0 ) )
17	4, 16	mpbird 167	. . . . . . 7 |- ( ph -> M # 0 )
18		lgsdilem2.5	. . . . . . . 8 |- ( ph -> N =/= 0 )
19		zapne 9669	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
20	10, 14, 19	sylancl 413	. . . . . . . 8 |- ( ph -> ( N # 0 <-> N =/= 0 ) )
21	18, 20	mpbird 167	. . . . . . 7 |- ( ph -> N # 0 )
22	12, 13, 17, 21	mulap0d 8949	. . . . . 6 |- ( ph -> ( M x. N ) # 0 )
23		zapne 9669	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
24	11, 14, 23	sylancl 413	. . . . . 6 |- ( ph -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
25	22, 24	mpbid 147	. . . . 5 |- ( ph -> ( M x. N ) =/= 0 )
26		nnabscl 11810	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
27	11, 25, 26	syl2anc 411	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) e. NN )
28	27	nnzd 9717	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
29	12	abscld 11891	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
30	13	abscld 11891	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
31	12	absge0d 11894	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
32		nnabscl 11810	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
33	10, 18, 32	syl2anc 411	. . . . . 6 |- ( ph -> ( abs ` N ) e. NN )
34	33	nnge1d 9297	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
35	29, 30, 31, 34	lemulge11d 9228	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
36	12, 13	absmuld 11904	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
37	35, 36	breqtrrd 4142	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
38		eluz2 9877	. . 3 |- ( ( abs ` ( M x. N ) ) e. ( ZZ>= ` ( abs ` M ) ) <-> ( ( abs ` M ) e. ZZ /\ ( abs ` ( M x. N ) ) e. ZZ /\ ( abs ` M ) <_ ( abs ` ( M x. N ) ) ) )
39	9, 28, 37, 38	syl3anbrc 1208	. 2 |- ( ph -> ( abs ` ( M x. N ) ) e. ( ZZ>= ` ( abs ` M ) ) )
40		1zzd 9621	. . . . 5 |- ( ph -> 1 e. ZZ )
41		lgsdilem2.1	. . . . . . 7 |- ( ph -> A e. ZZ )
42		lgsdilem2.6	. . . . . . . 8 |- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )
43	42	lgsfcl3 16020	. . . . . . 7 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
44	41, 3, 4, 43	syl3anc 1274	. . . . . 6 |- ( ph -> F : NN --> ZZ )
45	44	ffvelcdmda 5817	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. ZZ )
46		zmulcl 9648	. . . . . 6 |- ( ( k e. ZZ /\ v e. ZZ ) -> ( k x. v ) e. ZZ )
47	46	adantl 277	. . . . 5 |- ( ( ph /\ ( k e. ZZ /\ v e. ZZ ) ) -> ( k x. v ) e. ZZ )
48	7, 40, 45, 47	seqf 10850	. . . 4 |- ( ph -> seq 1 ( x. , F ) : NN --> ZZ )
49	48, 6	ffvelcdmd 5818	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. ZZ )
50	49	zcnd 9719	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
51		eleq1w 2295	. . . . 5 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
52		oveq2 6066	. . . . . 6 |- ( n = k -> ( A /L n ) = ( A /L k ) )
53		oveq1 6065	. . . . . 6 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
54	52, 53	oveq12d 6076	. . . . 5 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
55	51, 54	ifbieq1d 3649	. . . 4 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
56	6	peano2nnd 9269	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
57		elfzuz 10374	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
58		eluznn 9950	. . . . 5 |- ( ( ( ( abs ` M ) + 1 ) e. NN /\ k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) ) -> k e. NN )
59	56, 57, 58	syl2an 289	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> k e. NN )
60	41	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> A e. ZZ )
61		prmz 12833	. . . . . . . 8 |- ( k e. Prime -> k e. ZZ )
62	61	adantl 277	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. ZZ )
63		lgscl 16013	. . . . . . 7 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
64	60, 62, 63	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
65		simpr 110	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> k e. Prime )
66	3	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. ZZ )
67	4	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M =/= 0 )
68		pczcl 13021	. . . . . . 7 |- ( ( k e. Prime /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( k pCnt M ) e. NN0 )
69	65, 66, 67, 68	syl12anc 1272	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) e. NN0 )
70		zexpcl 10940	. . . . . 6 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt M ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt M ) ) e. ZZ )
71	64, 69, 70	syl2anc 411	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) e. ZZ )
72		1zzd 9621	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ -. k e. Prime ) -> 1 e. ZZ )
73		prmdc 12852	. . . . . 6 |- ( k e. NN -> DECID k e. Prime )
74	59, 73	syl 14	. . . . 5 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> DECID k e. Prime )
75	71, 72, 74	ifcldadc 3656	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. ZZ )
76	42, 55, 59, 75	fvmptd3 5776	. . 3 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
77		zq 9976	. . . . . . . . . 10 |- ( M e. ZZ -> M e. QQ )
78	66, 77	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> M e. QQ )
79		pcabs 13049	. . . . . . . . 9 |- ( ( k e. Prime /\ M e. QQ ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
80	65, 78, 79	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = ( k pCnt M ) )
81		elfzle1 10381	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
82	81	adantl 277	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) + 1 ) <_ k )
83		elfzelz 10378	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
84		zltp1le 9649	. . . . . . . . . . . . . 14 |- ( ( ( abs ` M ) e. ZZ /\ k e. ZZ ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
85	9, 83, 84	syl2an 289	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> ( ( abs ` M ) + 1 ) <_ k ) )
86	82, 85	mpbird 167	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( abs ` M ) < k )
87		zltnle 9640	. . . . . . . . . . . . 13 |- ( ( ( abs ` M ) e. ZZ /\ k e. ZZ ) -> ( ( abs ` M ) < k <-> -. k <_ ( abs ` M ) ) )
88	9, 83, 87	syl2an 289	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( ( abs ` M ) < k <-> -. k <_ ( abs ` M ) ) )
89	86, 88	mpbid 147	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> -. k <_ ( abs ` M ) )
90	89	adantr 276	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k <_ ( abs ` M ) )
91	66, 67, 5	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( abs ` M ) e. NN )
92		dvdsle 12555	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ ( abs ` M ) e. NN ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
93	62, 91, 92	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k || ( abs ` M ) -> k <_ ( abs ` M ) ) )
94	90, 93	mtod 669	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k || ( abs ` M ) )
95		pceq0 13045	. . . . . . . . . 10 |- ( ( k e. Prime /\ ( abs ` M ) e. NN ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
96	65, 91, 95	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( k pCnt ( abs ` M ) ) = 0 <-> -. k || ( abs ` M ) ) )
97	94, 96	mpbird 167	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt ( abs ` M ) ) = 0 )
98	80, 97	eqtr3d 2269	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
99	98	oveq2d 6074	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = ( ( A /L k ) ^ 0 ) )
100	64	zcnd 9719	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
101	100	exp0d 11054	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
102	99, 101	eqtrd 2267	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) = 1 )
103	102, 74	ifeq1dadc 3657	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = if ( k e. Prime , 1 , 1 ) )
104		ifiddc 3662	. . . . 5 |- (DECID k e. Prime -> if ( k e. Prime , 1 , 1 ) = 1 )
105	74, 104	syl 14	. . . 4 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , 1 , 1 ) = 1 )
106	103, 105	eqtrd 2267	. . 3 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = 1 )
107	76, 106	eqtrd 2267	. 2 |- ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) -> ( F ` k ) = 1 )
108	44	adantr 276	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> F : NN --> ZZ )
109		elnnuz 9909	. . . . . 6 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
110	109	biimpri 133	. . . . 5 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
111	110	adantl 277	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> k e. NN )
112	108, 111	ffvelcdmd 5818	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. ZZ )
113	112	zcnd 9719	. 2 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. CC )
114		mulcl 8270	. . 3 |- ( ( k e. CC /\ v e. CC ) -> ( k x. v ) e. CC )
115	114	adantl 277	. 2 |- ( ( ph /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
116	2, 8, 39, 50, 107, 113, 115	seq3id2 10912	1 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   # cap 8872   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   ...cfz 10361   seqcseq 10833   ^cexp 10924   abscabs 11707   || cdvds 12498   Primecprime 12829   pCnt cpc 13007   /Lclgs 15996

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-phi 12933  df-pc 13008  df-lgs 15997

This theorem is referenced by:  lgsdi  16036