Theorem lgsdilem2 15513

Description: Lemma for lgsdi 15514. (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 8069	. . 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 11411	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
6	3, 4, 5	syl2anc 411	. . 3 |- ( ph -> ( abs ` M ) e. NN )
7		nnuz 9684	. . 3 |- NN = ( ZZ>= ` 1 )
8	6, 7	eleqtrdi 2298	. 2 |- ( ph -> ( abs ` M ) e. ( ZZ>= ` 1 ) )
9	6	nnzd 9494	. . 3 |- ( ph -> ( abs ` M ) e. ZZ )
10		lgsdilem2.3	. . . . . 6 |- ( ph -> N e. ZZ )
11	3, 10	zmulcld 9501	. . . . 5 |- ( ph -> ( M x. N ) e. ZZ )
12	3	zcnd 9496	. . . . . . 7 |- ( ph -> M e. CC )
13	10	zcnd 9496	. . . . . . 7 |- ( ph -> N e. CC )
14		0z 9383	. . . . . . . . 9 |- 0 e. ZZ
15		zapne 9447	. . . . . . . . 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 9447	. . . . . . . . 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 8731	. . . . . 6 |- ( ph -> ( M x. N ) # 0 )
23		zapne 9447	. . . . . . 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 11411	. . . . 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 9494	. . 3 |- ( ph -> ( abs ` ( M x. N ) ) e. ZZ )
29	12	abscld 11492	. . . . 5 |- ( ph -> ( abs ` M ) e. RR )
30	13	abscld 11492	. . . . 5 |- ( ph -> ( abs ` N ) e. RR )
31	12	absge0d 11495	. . . . 5 |- ( ph -> 0 <_ ( abs ` M ) )
32		nnabscl 11411	. . . . . . 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 9079	. . . . 5 |- ( ph -> 1 <_ ( abs ` N ) )
35	29, 30, 31, 34	lemulge11d 9010	. . . 4 |- ( ph -> ( abs ` M ) <_ ( ( abs ` M ) x. ( abs ` N ) ) )
36	12, 13	absmuld 11505	. . . 4 |- ( ph -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
37	35, 36	breqtrrd 4072	. . 3 |- ( ph -> ( abs ` M ) <_ ( abs ` ( M x. N ) ) )
38		eluz2 9654	. . 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 9399	. . . . 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 15498	. . . . . . 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 5715	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. ZZ )
46		zmulcl 9426	. . . . . 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 10609	. . . 4 |- ( ph -> seq 1 ( x. , F ) : NN --> ZZ )
49	48, 6	ffvelcdmd 5716	. . 3 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. ZZ )
50	49	zcnd 9496	. 2 |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) e. CC )
51		eleq1w 2266	. . . . 5 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
52		oveq2 5952	. . . . . 6 |- ( n = k -> ( A /L n ) = ( A /L k ) )
53		oveq1 5951	. . . . . 6 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
54	52, 53	oveq12d 5962	. . . . 5 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
55	51, 54	ifbieq1d 3593	. . . 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 9051	. . . . 5 |- ( ph -> ( ( abs ` M ) + 1 ) e. NN )
57		elfzuz 10143	. . . . 5 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ( ZZ>= ` ( ( abs ` M ) + 1 ) ) )
58		eluznn 9721	. . . . 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 12433	. . . . . . . 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 15491	. . . . . . 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 12621	. . . . . . 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 10699	. . . . . 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 9399	. . . . 5 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ -. k e. Prime ) -> 1 e. ZZ )
73		prmdc 12452	. . . . . 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 3600	. . . 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 5673	. . 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 9747	. . . . . . . . . 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 12649	. . . . . . . . 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 10149	. . . . . . . . . . . . . 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 10147	. . . . . . . . . . . . . 14 |- ( k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) -> k e. ZZ )
84		zltp1le 9427	. . . . . . . . . . . . . 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 9418	. . . . . . . . . . . . 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 12155	. . . . . . . . . . 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 12645	. . . . . . . . . 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 2240	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( k pCnt M ) = 0 )
99	98	oveq2d 5960	. . . . . 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 9496	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
101	100	exp0d 10812	. . . . . 6 |- ( ( ( ph /\ k e. ( ( ( abs ` M ) + 1 ) ... ( abs ` ( M x. N ) ) ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ 0 ) = 1 )
102	99, 101	eqtrd 2238	. . . . 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 3601	. . . 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 3606	. . . . 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 2238	. . 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 2238	. 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 9685	. . . . . 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 5716	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. ZZ )
113	112	zcnd 9496	. 2 |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. CC )
114		mulcl 8052	. . 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 10671	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 2176   =/= wne 2376   ifcif 3571   class class class wbr 4044   |-> cmpt 4105   -->wf 5267   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   # cap 8654   NNcn 9036   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   QQcq 9740   ...cfz 10130   seqcseq 10592   ^cexp 10683   abscabs 11308   || cdvds 12098   Primecprime 12429   pCnt cpc 12607   /Lclgs 15474

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-2o 6503  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-proddc 11862  df-dvds 12099  df-gcd 12275  df-prm 12430  df-phi 12533  df-pc 12608  df-lgs 15475

This theorem is referenced by:  lgsdi  15514