Theorem fprodabs 12302

Description: The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)

Hypotheses

Ref	Expression
fprodabs.1	|- Z = ( ZZ>= ` M )
fprodabs.2	|- ( ph -> N e. Z )
fprodabs.3	|- ( ( ph /\ k e. Z ) -> A e. CC )

Assertion

Ref	Expression
fprodabs	|- ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) )

Distinct variable groups:   k, M   k, N   k, Z   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem fprodabs

Dummy variables a n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodabs.2	. . 3 |- ( ph -> N e. Z )
2		fprodabs.1	. . 3 |- Z = ( ZZ>= ` M )
3	1, 2	eleqtrdi 2325	. 2 |- ( ph -> N e. ( ZZ>= ` M ) )
4		oveq2 6058	. . . . . . 7 |- ( a = M -> ( M ... a ) = ( M ... M ) )
5	4	prodeq1d 12250	. . . . . 6 |- ( a = M -> prod_ k e. ( M ... a ) A = prod_ k e. ( M ... M ) A )
6	5	fveq2d 5674	. . . . 5 |- ( a = M -> ( abs ` prod_ k e. ( M ... a ) A ) = ( abs ` prod_ k e. ( M ... M ) A ) )
7	4	prodeq1d 12250	. . . . 5 |- ( a = M -> prod_ k e. ( M ... a ) ( abs ` A ) = prod_ k e. ( M ... M ) ( abs ` A ) )
8	6, 7	eqeq12d 2247	. . . 4 |- ( a = M -> ( ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) <-> ( abs ` prod_ k e. ( M ... M ) A ) = prod_ k e. ( M ... M ) ( abs ` A ) ) )
9	8	imbi2d 230	. . 3 |- ( a = M -> ( ( ph -> ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) ) <-> ( ph -> ( abs ` prod_ k e. ( M ... M ) A ) = prod_ k e. ( M ... M ) ( abs ` A ) ) ) )
10		oveq2 6058	. . . . . . 7 |- ( a = n -> ( M ... a ) = ( M ... n ) )
11	10	prodeq1d 12250	. . . . . 6 |- ( a = n -> prod_ k e. ( M ... a ) A = prod_ k e. ( M ... n ) A )
12	11	fveq2d 5674	. . . . 5 |- ( a = n -> ( abs ` prod_ k e. ( M ... a ) A ) = ( abs ` prod_ k e. ( M ... n ) A ) )
13	10	prodeq1d 12250	. . . . 5 |- ( a = n -> prod_ k e. ( M ... a ) ( abs ` A ) = prod_ k e. ( M ... n ) ( abs ` A ) )
14	12, 13	eqeq12d 2247	. . . 4 |- ( a = n -> ( ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) <-> ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) )
15	14	imbi2d 230	. . 3 |- ( a = n -> ( ( ph -> ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) ) <-> ( ph -> ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) ) )
16		oveq2 6058	. . . . . . 7 |- ( a = ( n + 1 ) -> ( M ... a ) = ( M ... ( n + 1 ) ) )
17	16	prodeq1d 12250	. . . . . 6 |- ( a = ( n + 1 ) -> prod_ k e. ( M ... a ) A = prod_ k e. ( M ... ( n + 1 ) ) A )
18	17	fveq2d 5674	. . . . 5 |- ( a = ( n + 1 ) -> ( abs ` prod_ k e. ( M ... a ) A ) = ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) )
19	16	prodeq1d 12250	. . . . 5 |- ( a = ( n + 1 ) -> prod_ k e. ( M ... a ) ( abs ` A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) )
20	18, 19	eqeq12d 2247	. . . 4 |- ( a = ( n + 1 ) -> ( ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) <-> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) ) )
21	20	imbi2d 230	. . 3 |- ( a = ( n + 1 ) -> ( ( ph -> ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) ) <-> ( ph -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) ) ) )
22		oveq2 6058	. . . . . . 7 |- ( a = N -> ( M ... a ) = ( M ... N ) )
23	22	prodeq1d 12250	. . . . . 6 |- ( a = N -> prod_ k e. ( M ... a ) A = prod_ k e. ( M ... N ) A )
24	23	fveq2d 5674	. . . . 5 |- ( a = N -> ( abs ` prod_ k e. ( M ... a ) A ) = ( abs ` prod_ k e. ( M ... N ) A ) )
25	22	prodeq1d 12250	. . . . 5 |- ( a = N -> prod_ k e. ( M ... a ) ( abs ` A ) = prod_ k e. ( M ... N ) ( abs ` A ) )
26	24, 25	eqeq12d 2247	. . . 4 |- ( a = N -> ( ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) <-> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) ) )
27	26	imbi2d 230	. . 3 |- ( a = N -> ( ( ph -> ( abs ` prod_ k e. ( M ... a ) A ) = prod_ k e. ( M ... a ) ( abs ` A ) ) <-> ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) ) ) )
28		csbfv2g 5711	. . . . . 6 |- ( M e. ZZ -> [_ M / k ]_ ( abs ` A ) = ( abs ` [_ M / k ]_ A ) )
29	28	adantl 277	. . . . 5 |- ( ( ph /\ M e. ZZ ) -> [_ M / k ]_ ( abs ` A ) = ( abs ` [_ M / k ]_ A ) )
30		fzsn 10400	. . . . . . . 8 |- ( M e. ZZ -> ( M ... M ) = { M } )
31	30	adantl 277	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> ( M ... M ) = { M } )
32	31	prodeq1d 12250	. . . . . 6 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. ( M ... M ) ( abs ` A ) = prod_ k e. { M } ( abs ` A ) )
33		simpr 110	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> M e. ZZ )
34		uzid 9868	. . . . . . . . . . . 12 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
35	34, 2	eleqtrrdi 2326	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. Z )
36		fprodabs.3	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. Z ) -> A e. CC )
37	36	ralrimiva 2615	. . . . . . . . . . . 12 |- ( ph -> A. k e. Z A e. CC )
38		nfcsb1v 3171	. . . . . . . . . . . . . 14 |- F/_ k [_ M / k ]_ A
39	38	nfel1 2395	. . . . . . . . . . . . 13 |- F/ k [_ M / k ]_ A e. CC
40		csbeq1a 3147	. . . . . . . . . . . . . 14 |- ( k = M -> A = [_ M / k ]_ A )
41	40	eleq1d 2301	. . . . . . . . . . . . 13 |- ( k = M -> ( A e. CC <-> [_ M / k ]_ A e. CC ) )
42	39, 41	rspc 2915	. . . . . . . . . . . 12 |- ( M e. Z -> ( A. k e. Z A e. CC -> [_ M / k ]_ A e. CC ) )
43	37, 42	mpan9 281	. . . . . . . . . . 11 |- ( ( ph /\ M e. Z ) -> [_ M / k ]_ A e. CC )
44	35, 43	sylan2 286	. . . . . . . . . 10 |- ( ( ph /\ M e. ZZ ) -> [_ M / k ]_ A e. CC )
45	44	abscld 11866	. . . . . . . . 9 |- ( ( ph /\ M e. ZZ ) -> ( abs ` [_ M / k ]_ A ) e. RR )
46	45	recnd 8302	. . . . . . . 8 |- ( ( ph /\ M e. ZZ ) -> ( abs ` [_ M / k ]_ A ) e. CC )
47	29, 46	eqeltrd 2309	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> [_ M / k ]_ ( abs ` A ) e. CC )
48		prodsns 12289	. . . . . . 7 |- ( ( M e. ZZ /\ [_ M / k ]_ ( abs ` A ) e. CC ) -> prod_ k e. { M } ( abs ` A ) = [_ M / k ]_ ( abs ` A ) )
49	33, 47, 48	syl2anc 411	. . . . . 6 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. { M } ( abs ` A ) = [_ M / k ]_ ( abs ` A ) )
50	32, 49	eqtrd 2265	. . . . 5 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. ( M ... M ) ( abs ` A ) = [_ M / k ]_ ( abs ` A ) )
51	30	prodeq1d 12250	. . . . . . . 8 |- ( M e. ZZ -> prod_ k e. ( M ... M ) A = prod_ k e. { M } A )
52	51	adantl 277	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. ( M ... M ) A = prod_ k e. { M } A )
53		prodsns 12289	. . . . . . . 8 |- ( ( M e. ZZ /\ [_ M / k ]_ A e. CC ) -> prod_ k e. { M } A = [_ M / k ]_ A )
54	33, 44, 53	syl2anc 411	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. { M } A = [_ M / k ]_ A )
55	52, 54	eqtrd 2265	. . . . . 6 |- ( ( ph /\ M e. ZZ ) -> prod_ k e. ( M ... M ) A = [_ M / k ]_ A )
56	55	fveq2d 5674	. . . . 5 |- ( ( ph /\ M e. ZZ ) -> ( abs ` prod_ k e. ( M ... M ) A ) = ( abs ` [_ M / k ]_ A ) )
57	29, 50, 56	3eqtr4rd 2276	. . . 4 |- ( ( ph /\ M e. ZZ ) -> ( abs ` prod_ k e. ( M ... M ) A ) = prod_ k e. ( M ... M ) ( abs ` A ) )
58	57	expcom 116	. . 3 |- ( M e. ZZ -> ( ph -> ( abs ` prod_ k e. ( M ... M ) A ) = prod_ k e. ( M ... M ) ( abs ` A ) ) )
59		simp3 1026	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) )
60		peano2uz 9915	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
61		csbfv2g 5711	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( ZZ>= ` M ) -> [_ ( n + 1 ) / k ]_ ( abs ` A ) = ( abs ` [_ ( n + 1 ) / k ]_ A ) )
62	60, 61	syl 14	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` M ) -> [_ ( n + 1 ) / k ]_ ( abs ` A ) = ( abs ` [_ ( n + 1 ) / k ]_ A ) )
63	62	eqcomd 2238	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( abs ` [_ ( n + 1 ) / k ]_ A ) = [_ ( n + 1 ) / k ]_ ( abs ` A ) )
64	63	3ad2ant2 1046	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( abs ` [_ ( n + 1 ) / k ]_ A ) = [_ ( n + 1 ) / k ]_ ( abs ` A ) )
65	59, 64	oveq12d 6068	. . . . . . 7 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( ( abs ` prod_ k e. ( M ... n ) A ) x. ( abs ` [_ ( n + 1 ) / k ]_ A ) ) = ( prod_ k e. ( M ... n ) ( abs ` A ) x. [_ ( n + 1 ) / k ]_ ( abs ` A ) ) )
66		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
67		elfzuz 10355	. . . . . . . . . . . . . 14 |- ( k e. ( M ... ( n + 1 ) ) -> k e. ( ZZ>= ` M ) )
68	67, 2	eleqtrrdi 2326	. . . . . . . . . . . . 13 |- ( k e. ( M ... ( n + 1 ) ) -> k e. Z )
69	68, 36	sylan2 286	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( M ... ( n + 1 ) ) ) -> A e. CC )
70	69	adantlr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` M ) ) /\ k e. ( M ... ( n + 1 ) ) ) -> A e. CC )
71	66, 70	fprodp1s 12288	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> prod_ k e. ( M ... ( n + 1 ) ) A = ( prod_ k e. ( M ... n ) A x. [_ ( n + 1 ) / k ]_ A ) )
72	71	fveq2d 5674	. . . . . . . . 9 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = ( abs ` ( prod_ k e. ( M ... n ) A x. [_ ( n + 1 ) / k ]_ A ) ) )
73		eluzel2 9858	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` M ) -> M e. ZZ )
74	73	adantl 277	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> M e. ZZ )
75		eluzelz 9863	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
76	75	adantl 277	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> n e. ZZ )
77	74, 76	fzfigd 10793	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( M ... n ) e. Fin )
78		elfzuz 10355	. . . . . . . . . . . . . 14 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
79	78, 2	eleqtrrdi 2326	. . . . . . . . . . . . 13 |- ( k e. ( M ... n ) -> k e. Z )
80	79, 36	sylan2 286	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( M ... n ) ) -> A e. CC )
81	80	adantlr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` M ) ) /\ k e. ( M ... n ) ) -> A e. CC )
82	77, 81	fprodcl 12293	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> prod_ k e. ( M ... n ) A e. CC )
83	60, 2	eleqtrrdi 2326	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. Z )
84		nfcsb1v 3171	. . . . . . . . . . . . . 14 |- F/_ k [_ ( n + 1 ) / k ]_ A
85	84	nfel1 2395	. . . . . . . . . . . . 13 |- F/ k [_ ( n + 1 ) / k ]_ A e. CC
86		csbeq1a 3147	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> A = [_ ( n + 1 ) / k ]_ A )
87	86	eleq1d 2301	. . . . . . . . . . . . 13 |- ( k = ( n + 1 ) -> ( A e. CC <-> [_ ( n + 1 ) / k ]_ A e. CC ) )
88	85, 87	rspc 2915	. . . . . . . . . . . 12 |- ( ( n + 1 ) e. Z -> ( A. k e. Z A e. CC -> [_ ( n + 1 ) / k ]_ A e. CC ) )
89	37, 88	mpan9 281	. . . . . . . . . . 11 |- ( ( ph /\ ( n + 1 ) e. Z ) -> [_ ( n + 1 ) / k ]_ A e. CC )
90	83, 89	sylan2 286	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> [_ ( n + 1 ) / k ]_ A e. CC )
91	82, 90	absmuld 11879	. . . . . . . . 9 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( abs ` ( prod_ k e. ( M ... n ) A x. [_ ( n + 1 ) / k ]_ A ) ) = ( ( abs ` prod_ k e. ( M ... n ) A ) x. ( abs ` [_ ( n + 1 ) / k ]_ A ) ) )
92	72, 91	eqtrd 2265	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = ( ( abs ` prod_ k e. ( M ... n ) A ) x. ( abs ` [_ ( n + 1 ) / k ]_ A ) ) )
93	92	3adant3 1044	. . . . . . 7 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = ( ( abs ` prod_ k e. ( M ... n ) A ) x. ( abs ` [_ ( n + 1 ) / k ]_ A ) ) )
94	70	abscld 11866	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( ZZ>= ` M ) ) /\ k e. ( M ... ( n + 1 ) ) ) -> ( abs ` A ) e. RR )
95	94	recnd 8302	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( ZZ>= ` M ) ) /\ k e. ( M ... ( n + 1 ) ) ) -> ( abs ` A ) e. CC )
96	66, 95	fprodp1s 12288	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) = ( prod_ k e. ( M ... n ) ( abs ` A ) x. [_ ( n + 1 ) / k ]_ ( abs ` A ) ) )
97	96	3adant3 1044	. . . . . . 7 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) = ( prod_ k e. ( M ... n ) ( abs ` A ) x. [_ ( n + 1 ) / k ]_ ( abs ` A ) ) )
98	65, 93, 97	3eqtr4d 2275	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) /\ ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) )
99	98	3exp 1229	. . . . 5 |- ( ph -> ( n e. ( ZZ>= ` M ) -> ( ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) ) ) )
100	99	com12 30	. . . 4 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) ) ) )
101	100	a2d 26	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ( ph -> ( abs ` prod_ k e. ( M ... n ) A ) = prod_ k e. ( M ... n ) ( abs ` A ) ) -> ( ph -> ( abs ` prod_ k e. ( M ... ( n + 1 ) ) A ) = prod_ k e. ( M ... ( n + 1 ) ) ( abs ` A ) ) ) )
102	9, 15, 21, 27, 58, 101	uzind4 9920	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) ) )
103	3, 102	mpcom 36	1 |- ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   [_csb 3138   {csn 3689   ` cfv 5352  (class class class)co 6050   CCcc 8125   1c1 8128   + caddc 8130   x. cmul 8132   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   abscabs 11682   prod_cprod 12236

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237

This theorem is referenced by: (None)