Theorem iprodefisumlem 31626

Description: Lemma for iprodefisum 31627. (Contributed by Scott Fenton, 11-Feb-2018.)

Hypotheses

Ref	Expression
iprodefisumlem.1	|- Z = ( ZZ>= ` M )
iprodefisumlem.2	|- ( ph -> M e. ZZ )
iprodefisumlem.3	|- ( ph -> F : Z --> CC )

Assertion

Ref	Expression
iprodefisumlem	|- ( ph -> seq M ( x. , ( exp o. F ) ) = ( exp o. seq M ( + , F ) ) )

Proof of Theorem iprodefisumlem

Dummy variables j k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodefisumlem.1	. . . 4 |- Z = ( ZZ>= ` M )
2		iprodefisumlem.2	. . . 4 |- ( ph -> M e. ZZ )
3		iprodefisumlem.3	. . . . . 6 |- ( ph -> F : Z --> CC )
4		fvco3 6275	. . . . . 6 |- ( ( F : Z --> CC /\ k e. Z ) -> ( ( exp o. F ) ` k ) = ( exp ` ( F ` k ) ) )
5	3, 4	sylan 488	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( exp o. F ) ` k ) = ( exp ` ( F ` k ) ) )
6	3	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
7		efcl 14813	. . . . . 6 |- ( ( F ` k ) e. CC -> ( exp ` ( F ` k ) ) e. CC )
8	6, 7	syl 17	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( exp ` ( F ` k ) ) e. CC )
9	5, 8	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( exp o. F ) ` k ) e. CC )
10	1, 2, 9	prodf 14619	. . 3 |- ( ph -> seq M ( x. , ( exp o. F ) ) : Z --> CC )
11		ffn 6045	. . 3 |- ( seq M ( x. , ( exp o. F ) ) : Z --> CC -> seq M ( x. , ( exp o. F ) ) Fn Z )
12	10, 11	syl 17	. 2 |- ( ph -> seq M ( x. , ( exp o. F ) ) Fn Z )
13		eff 14812	. . . 4 |- exp : CC --> CC
14		ffn 6045	. . . 4 |- ( exp : CC --> CC -> exp Fn CC )
15	13, 14	ax-mp 5	. . 3 |- exp Fn CC
16	1, 2, 6	serf 12829	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
17		fnfco 6069	. . 3 |- ( ( exp Fn CC /\ seq M ( + , F ) : Z --> CC ) -> ( exp o. seq M ( + , F ) ) Fn Z )
18	15, 16, 17	sylancr 695	. 2 |- ( ph -> ( exp o. seq M ( + , F ) ) Fn Z )
19		fveq2 6191	. . . . . . . 8 |- ( j = M -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( seq M ( x. , ( exp o. F ) ) ` M ) )
20		fveq2 6191	. . . . . . . . 9 |- ( j = M -> ( seq M ( + , F ) ` j ) = ( seq M ( + , F ) ` M ) )
21	20	fveq2d 6195	. . . . . . . 8 |- ( j = M -> ( exp ` ( seq M ( + , F ) ` j ) ) = ( exp ` ( seq M ( + , F ) ` M ) ) )
22	19, 21	eqeq12d 2637	. . . . . . 7 |- ( j = M -> ( ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) <-> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( exp ` ( seq M ( + , F ) ` M ) ) ) )
23	22	imbi2d 330	. . . . . 6 |- ( j = M -> ( ( ph -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) ) <-> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( exp ` ( seq M ( + , F ) ` M ) ) ) ) )
24		fveq2 6191	. . . . . . . 8 |- ( j = n -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( seq M ( x. , ( exp o. F ) ) ` n ) )
25		fveq2 6191	. . . . . . . . 9 |- ( j = n -> ( seq M ( + , F ) ` j ) = ( seq M ( + , F ) ` n ) )
26	25	fveq2d 6195	. . . . . . . 8 |- ( j = n -> ( exp ` ( seq M ( + , F ) ` j ) ) = ( exp ` ( seq M ( + , F ) ` n ) ) )
27	24, 26	eqeq12d 2637	. . . . . . 7 |- ( j = n -> ( ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) <-> ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) )
28	27	imbi2d 330	. . . . . 6 |- ( j = n -> ( ( ph -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) ) <-> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) ) )
29		fveq2 6191	. . . . . . . 8 |- ( j = ( n + 1 ) -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) )
30		fveq2 6191	. . . . . . . . 9 |- ( j = ( n + 1 ) -> ( seq M ( + , F ) ` j ) = ( seq M ( + , F ) ` ( n + 1 ) ) )
31	30	fveq2d 6195	. . . . . . . 8 |- ( j = ( n + 1 ) -> ( exp ` ( seq M ( + , F ) ` j ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) )
32	29, 31	eqeq12d 2637	. . . . . . 7 |- ( j = ( n + 1 ) -> ( ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) <-> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) ) )
33	32	imbi2d 330	. . . . . 6 |- ( j = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) ) <-> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) ) ) )
34		fveq2 6191	. . . . . . . 8 |- ( j = k -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( seq M ( x. , ( exp o. F ) ) ` k ) )
35		fveq2 6191	. . . . . . . . 9 |- ( j = k -> ( seq M ( + , F ) ` j ) = ( seq M ( + , F ) ` k ) )
36	35	fveq2d 6195	. . . . . . . 8 |- ( j = k -> ( exp ` ( seq M ( + , F ) ` j ) ) = ( exp ` ( seq M ( + , F ) ` k ) ) )
37	34, 36	eqeq12d 2637	. . . . . . 7 |- ( j = k -> ( ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) <-> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) ) )
38	37	imbi2d 330	. . . . . 6 |- ( j = k -> ( ( ph -> ( seq M ( x. , ( exp o. F ) ) ` j ) = ( exp ` ( seq M ( + , F ) ` j ) ) ) <-> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) ) ) )
39		uzid 11702	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
40	2, 39	syl 17	. . . . . . . . . 10 |- ( ph -> M e. ( ZZ>= ` M ) )
41	40, 1	syl6eleqr 2712	. . . . . . . . 9 |- ( ph -> M e. Z )
42		fvco3 6275	. . . . . . . . 9 |- ( ( F : Z --> CC /\ M e. Z ) -> ( ( exp o. F ) ` M ) = ( exp ` ( F ` M ) ) )
43	3, 41, 42	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( exp o. F ) ` M ) = ( exp ` ( F ` M ) ) )
44		seq1 12814	. . . . . . . . 9 |- ( M e. ZZ -> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( ( exp o. F ) ` M ) )
45	2, 44	syl 17	. . . . . . . 8 |- ( ph -> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( ( exp o. F ) ` M ) )
46		seq1 12814	. . . . . . . . . 10 |- ( M e. ZZ -> ( seq M ( + , F ) ` M ) = ( F ` M ) )
47	2, 46	syl 17	. . . . . . . . 9 |- ( ph -> ( seq M ( + , F ) ` M ) = ( F ` M ) )
48	47	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( exp ` ( seq M ( + , F ) ` M ) ) = ( exp ` ( F ` M ) ) )
49	43, 45, 48	3eqtr4d 2666	. . . . . . 7 |- ( ph -> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( exp ` ( seq M ( + , F ) ` M ) ) )
50	49	a1i 11	. . . . . 6 |- ( M e. ZZ -> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` M ) = ( exp ` ( seq M ( + , F ) ` M ) ) ) )
51		oveq1 6657	. . . . . . . . . . 11 |- ( ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) -> ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) )
52	51	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) )
53	3	adantl 482	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> F : Z --> CC )
54		peano2uz 11741	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
55	54, 1	syl6eleqr 2712	. . . . . . . . . . . . . . 15 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. Z )
56	55	adantr 481	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( n + 1 ) e. Z )
57		fvco3 6275	. . . . . . . . . . . . . 14 |- ( ( F : Z --> CC /\ ( n + 1 ) e. Z ) -> ( ( exp o. F ) ` ( n + 1 ) ) = ( exp ` ( F ` ( n + 1 ) ) ) )
58	53, 56, 57	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( ( exp o. F ) ` ( n + 1 ) ) = ( exp ` ( F ` ( n + 1 ) ) ) )
59	58	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( exp ` ( F ` ( n + 1 ) ) ) ) )
60	16	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , F ) ` n ) e. CC )
61	60	expcom 451	. . . . . . . . . . . . . . 15 |- ( n e. Z -> ( ph -> ( seq M ( + , F ) ` n ) e. CC ) )
62	1	eqcomi 2631	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` M ) = Z
63	61, 62	eleq2s 2719	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( + , F ) ` n ) e. CC ) )
64	63	imp 445	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( seq M ( + , F ) ` n ) e. CC )
65	53, 56	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( F ` ( n + 1 ) ) e. CC )
66		efadd 14824	. . . . . . . . . . . . 13 |- ( ( ( seq M ( + , F ) ` n ) e. CC /\ ( F ` ( n + 1 ) ) e. CC ) -> ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) = ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( exp ` ( F ` ( n + 1 ) ) ) ) )
67	64, 65, 66	syl2anc 693	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) = ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( exp ` ( F ` ( n + 1 ) ) ) ) )
68	59, 67	eqtr4d 2659	. . . . . . . . . . 11 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) )
69	68	3adant3 1081	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( ( exp ` ( seq M ( + , F ) ` n ) ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) )
70	52, 69	eqtrd 2656	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) = ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) )
71		seqp1 12816	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) )
72	71	adantr 481	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) )
73	72	3adant3 1081	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( ( seq M ( x. , ( exp o. F ) ) ` n ) x. ( ( exp o. F ) ` ( n + 1 ) ) ) )
74		seqp1 12816	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( + , F ) ` ( n + 1 ) ) = ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) )
75	74	adantr 481	. . . . . . . . . . 11 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( seq M ( + , F ) ` ( n + 1 ) ) = ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) )
76	75	fveq2d 6195	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` M ) /\ ph ) -> ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) = ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) )
77	76	3adant3 1081	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) = ( exp ` ( ( seq M ( + , F ) ` n ) + ( F ` ( n + 1 ) ) ) ) )
78	70, 73, 77	3eqtr4d 2666	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` M ) /\ ph /\ ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) )
79	78	3exp 1264	. . . . . . 7 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) ) ) )
80	79	a2d 29	. . . . . 6 |- ( n e. ( ZZ>= ` M ) -> ( ( ph -> ( seq M ( x. , ( exp o. F ) ) ` n ) = ( exp ` ( seq M ( + , F ) ` n ) ) ) -> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` ( n + 1 ) ) = ( exp ` ( seq M ( + , F ) ` ( n + 1 ) ) ) ) ) )
81	23, 28, 33, 38, 50, 80	uzind4 11746	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) ) )
82	81, 1	eleq2s 2719	. . . 4 |- ( k e. Z -> ( ph -> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) ) )
83	82	impcom 446	. . 3 |- ( ( ph /\ k e. Z ) -> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) )
84		fvco3 6275	. . . 4 |- ( ( seq M ( + , F ) : Z --> CC /\ k e. Z ) -> ( ( exp o. seq M ( + , F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) )
85	16, 84	sylan 488	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( exp o. seq M ( + , F ) ) ` k ) = ( exp ` ( seq M ( + , F ) ` k ) ) )
86	83, 85	eqtr4d 2659	. 2 |- ( ( ph /\ k e. Z ) -> ( seq M ( x. , ( exp o. F ) ) ` k ) = ( ( exp o. seq M ( + , F ) ) ` k ) )
87	12, 18, 86	eqfnfvd 6314	1 |- ( ph -> seq M ( x. , ( exp o. F ) ) = ( exp o. seq M ( + , F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   expce 14792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798

This theorem is referenced by:  iprodefisum  31627