Theorem lgsdilem2 15628

Description: Lemma for lgsdi 15629. (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 8104	. . 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 11526	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 411	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 9719	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	eleqtrdi 2300	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 9529	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 9536	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 9531	. . . . . . 7 |- ( ph -> M e. CC )
13	10	zcnd 9531	. . . . . . 7 |- ( ph -> N e. CC )
14		0z 9418	. . . . . . . . 9 |- 0 e. ZZ
15		zapne 9482	. . . . . . . . 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 9482	. . . . . . . . 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 8766	. . . . . 6 |- ( ph -> ( M x. N ) # 0 )
23		zapne 9482	. . . . . . 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 11526	. . . . 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 9529	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
29	12	abscld 11607	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
30	13	abscld 11607	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
31	12	absge0d 11610	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
32		nnabscl 11526	. . . . . . 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 9114	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
35	29, 30, 31, 34	lemulge11d 9045	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
36	12, 13	absmuld 11620	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
37	35, 36	breqtrrd 4087	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
38		eluz2 9689	. . 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 1184	. 2 |- ( ph -> ( abs ` ( M x. N ) ) e. ( ZZ>= ` ( abs ` M ) ) )
40		1zzd 9434	. . . . 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 15613	. . . . . . 7 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> F : NN --> ZZ )
44	41, 3, 4, 43	syl3anc 1250	. . . . . 6 |- ( ph -> F : NN --> ZZ )
45	44	ffvelcdmda 5738	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. ZZ )
46		zmulcl 9461	. . . . . 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 10646	. . . 4 |- ( ph -> seq 1 ( x. , F ) : NN --> ZZ )
49	48, 6	ffvelcdmd 5739	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. ZZ )
50	49	zcnd 9531	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
51		eleq1w 2268	. . . . 5 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
52		oveq2 5975	. . . . . 6 |- ( n = k -> ( A /L n ) = ( A /L k ) )
53		oveq1 5974	. . . . . 6 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
54	52, 53	oveq12d 5985	. . . . 5 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
55	51, 54	ifbieq1d 3602	. . . 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 9086	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
57		elfzuz 10178	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
58		eluznn 9756	. . . . 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 12548	. . . . . . . 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 15606	. . . . . . 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 12736	. . . . . . 7 |- ( ( k e. Prime /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( k pCnt M ) e. NN0 )
69	65, 66, 67, 68	syl12anc 1248	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) e. NN0 )
70		zexpcl 10736	. . . . . 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 9434	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ -. k e. Prime ) -> 1 e. ZZ )
73		prmdc 12567	. . . . . 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 3609	. . . 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 5696	. . 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 9782	. . . . . . . . . 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 12764	. . . . . . . . 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 10184	. . . . . . . . . . . . . 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 10182	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
84		zltp1le 9462	. . . . . . . . . . . . . 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 9453	. . . . . . . . . . . . 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 12270	. . . . . . . . . . 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 665	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> -. k || ( abs ` M ) )
95		pceq0 12760	. . . . . . . . . 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 2242	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
99	98	oveq2d 5983	. . . . . 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 9531	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
101	100	exp0d 10849	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
102	99, 101	eqtrd 2240	. . . . 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 3610	. . . 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 3615	. . . . 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 2240	. . 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 2240	. 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 9720	. . . . . 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 5739	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. ZZ )
113	112	zcnd 9531	. 2 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. CC )
114		mulcl 8087	. . 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 10708	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 836   = wceq 1373   e. wcel 2178   =/= wne 2378   ifcif 3579   class class class wbr 4059   |-> cmpt 4121   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   # cap 8689   NNcn 9071   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   QQcq 9775   ...cfz 10165   seqcseq 10629   ^cexp 10720   abscabs 11423   || cdvds 12213   Primecprime 12544   pCnt cpc 12722   /Lclgs 15589

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-2o 6526  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-proddc 11977  df-dvds 12214  df-gcd 12390  df-prm 12545  df-phi 12648  df-pc 12723  df-lgs 15590

This theorem is referenced by:  lgsdi  15629