Theorem iseqfveq2 9448

Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)

Hypotheses

Ref	Expression
iseqfveq2.1	|- ( ph -> K e. ( ZZ>= ` M ) )
iseqfveq2.2	|- ( ph -> ( seq M ( .+ , F , S ) ` K ) = ( G ` K ) )
iseqfveq2.s	|- ( ph -> S e. V )
iseqfveq2.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqfveq2.g	|- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
iseqfveq2.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
iseqfveq2.3	|- ( ph -> N e. ( ZZ>= ` K ) )
iseqfveq2.4	|- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
iseqfveq2	|- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) )

Distinct variable groups:   x, k, y, F   k, G, x, y   k, K, x, y   k, N, x, y   ph, k, x, y   k, M, x, y   .+ , k, x, y   S, k, x, y

Allowed substitution hints:   V( x, y, k)

Proof of Theorem iseqfveq2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqfveq2.3	. . 3 |- ( ph -> N e. ( ZZ>= ` K ) )
2		eluzfz2 9051	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2141	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5198	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` K ) )
6		fveq2 5198	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` K ) )
7	5, 6	eqeq12d 2095	. . . . . 6 |- ( z = K -> ( ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) <-> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) ) )
8	4, 7	imbi12d 232	. . . . 5 |- ( z = K -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) <-> ( K e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) ) ) )
9	8	imbi2d 228	. . . 4 |- ( z = K -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) ) <-> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) ) ) ) )
10		eleq1 2141	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5198	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` w ) )
12		fveq2 5198	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` w ) )
13	11, 12	eqeq12d 2095	. . . . . 6 |- ( z = w -> ( ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) <-> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) )
14	10, 13	imbi12d 232	. . . . 5 |- ( z = w -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) <-> ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) ) )
15	14	imbi2d 228	. . . 4 |- ( z = w -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) ) <-> ( ph -> ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) ) ) )
16		eleq1 2141	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5198	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` ( w + 1 ) ) )
18		fveq2 5198	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2095	. . . . . 6 |- ( z = ( w + 1 ) -> ( ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) <-> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) )
20	16, 19	imbi12d 232	. . . . 5 |- ( z = ( w + 1 ) -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) <-> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) )
21	20	imbi2d 228	. . . 4 |- ( z = ( w + 1 ) -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) ) <-> ( ph -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) ) )
22		eleq1 2141	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5198	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` N ) )
24		fveq2 5198	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` N ) )
25	23, 24	eqeq12d 2095	. . . . . 6 |- ( z = N -> ( ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) <-> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) )
26	22, 25	imbi12d 232	. . . . 5 |- ( z = N -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) <-> ( N e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) ) )
27	26	imbi2d 228	. . . 4 |- ( z = N -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq K ( .+ , G , S ) ` z ) ) ) <-> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) ) ) )
28		iseqfveq2.2	. . . . . . 7 |- ( ph -> ( seq M ( .+ , F , S ) ` K ) = ( G ` K ) )
29		iseqfveq2.1	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
30		eluzelz 8628	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
31	29, 30	syl 14	. . . . . . . 8 |- ( ph -> K e. ZZ )
32		iseqfveq2.s	. . . . . . . 8 |- ( ph -> S e. V )
33		iseqfveq2.g	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
34		iseqfveq2.pl	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
35	31, 32, 33, 34	iseq1 9442	. . . . . . 7 |- ( ph -> ( seq K ( .+ , G , S ) ` K ) = ( G ` K ) )
36	28, 35	eqtr4d 2116	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) )
37	36	a1d 22	. . . . 5 |- ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) ) )
38	37	a1i 9	. . . 4 |- ( K e. ZZ -> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` K ) = ( seq K ( .+ , G , S ) ` K ) ) ) )
39		peano2fzr 9056	. . . . . . . . . 10 |- ( ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) -> w e. ( K ... N ) )
40	39	adantl 271	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( K ... N ) )
41	40	expr 367	. . . . . . . 8 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w + 1 ) e. ( K ... N ) -> w e. ( K ... N ) ) )
42	41	imim1d 74	. . . . . . 7 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) ) )
43		oveq1 5539	. . . . . . . . . 10 |- ( ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) -> ( ( seq M ( .+ , F , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
44		simprl 497	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` K ) )
45	29	adantr 270	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
46		uztrn 8635	. . . . . . . . . . . . 13 |- ( ( w e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> w e. ( ZZ>= ` M ) )
47	44, 45, 46	syl2anc 403	. . . . . . . . . . . 12 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` M ) )
48	32	adantr 270	. . . . . . . . . . . 12 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> S e. V )
49		iseqfveq2.f	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
50	49	adantlr 460	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
51	34	adantlr 460	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
52	47, 48, 50, 51	iseqp1 9445	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( ( seq M ( .+ , F , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
53	33	adantlr 460	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
54	44, 48, 53, 51	iseqp1 9445	. . . . . . . . . . . 12 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G , S ) ` ( w + 1 ) ) = ( ( seq K ( .+ , G , S ) ` w ) .+ ( G ` ( w + 1 ) ) ) )
55		eluzp1p1 8644	. . . . . . . . . . . . . . . 16 |- ( w e. ( ZZ>= ` K ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
56	55	ad2antrl 473	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
57		elfzuz3 9042	. . . . . . . . . . . . . . . 16 |- ( ( w + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
58	57	ad2antll 474	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
59		elfzuzb 9039	. . . . . . . . . . . . . . 15 |- ( ( w + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( w + 1 ) ) ) )
60	56, 58, 59	sylanbrc 408	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ( K + 1 ) ... N ) )
61		iseqfveq2.4	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )
62	61	ralrimiva 2434	. . . . . . . . . . . . . . 15 |- ( ph -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
63	62	adantr 270	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
64		fveq2 5198	. . . . . . . . . . . . . . . 16 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
65		fveq2 5198	. . . . . . . . . . . . . . . 16 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
66	64, 65	eqeq12d 2095	. . . . . . . . . . . . . . 15 |- ( k = ( w + 1 ) -> ( ( F ` k ) = ( G ` k ) <-> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) ) )
67	66	rspcv 2697	. . . . . . . . . . . . . 14 |- ( ( w + 1 ) e. ( ( K + 1 ) ... N ) -> ( A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) ) )
68	60, 63, 67	sylc 61	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
69	68	oveq2d 5548	. . . . . . . . . . . 12 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq K ( .+ , G , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G , S ) ` w ) .+ ( G ` ( w + 1 ) ) ) )
70	54, 69	eqtr4d 2116	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G , S ) ` ( w + 1 ) ) = ( ( seq K ( .+ , G , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
71	52, 70	eqeq12d 2095	. . . . . . . . . 10 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) <-> ( ( seq M ( .+ , F , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G , S ) ` w ) .+ ( F ` ( w + 1 ) ) ) ) )
72	43, 71	syl5ibr 154	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) )
73	72	expr 367	. . . . . . . 8 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) )
74	73	a2d 26	. . . . . . 7 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) )
75	42, 74	syld 44	. . . . . 6 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) )
76	75	expcom 114	. . . . 5 |- ( w e. ( ZZ>= ` K ) -> ( ph -> ( ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) ) )
77	76	a2d 26	. . . 4 |- ( w e. ( ZZ>= ` K ) -> ( ( ph -> ( w e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq K ( .+ , G , S ) ` w ) ) ) -> ( ph -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` ( w + 1 ) ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) ) ) ) )
78	9, 15, 21, 27, 38, 77	uzind4 8676	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) ) )
79	1, 78	mpcom 36	. 2 |- ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) )
80	3, 79	mpd 13	1 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-iseq 9432

This theorem is referenced by:  iseqfeq2  9449  iseqfveq  9450