Theorem iseqfveq2 9228

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 8896	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2100	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5178	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` K ) )
6		fveq2 5178	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` K ) )
7	5, 6	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5178	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` w ) )
12		fveq2 5178	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` w ) )
13	11, 12	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5178	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` ( w + 1 ) ) )
18		fveq2 5178	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5178	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , F , S ) ` N ) )
24		fveq2 5178	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G , S ) ` z ) = ( seq K ( .+ , G , S ) ` N ) )
25	23, 24	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 8482	. . . . . . . . 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 9222	. . . . . . 7 |- ( ph -> ( seq K ( .+ , G , S ) ` K ) = ( G ` K ) )
36	28, 35	eqtr4d 2075	. . . . . 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 8901	. . . . . . . . . 10 |- ( ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) -> w e. ( K ... N ) )
40	39	adantl 262	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( K ... N ) )
41	40	expr 357	. . . . . . . 8 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w + 1 ) e. ( K ... N ) -> w e. ( K ... N ) ) )
42	41	imim1d 69	. . . . . . 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 5519	. . . . . . . . . 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 483	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` K ) )
45	29	adantr 261	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
46		uztrn 8489	. . . . . . . . . . . . 13 |- ( ( w e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> w e. ( ZZ>= ` M ) )
47	44, 45, 46	syl2anc 391	. . . . . . . . . . . 12 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` M ) )
48	32	adantr 261	. . . . . . . . . . . 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 446	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
51	34	adantlr 446	. . . . . . . . . . . 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 9225	. . . . . . . . . . 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 446	. . . . . . . . . . . . 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 9225	. . . . . . . . . . . 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 8498	. . . . . . . . . . . . . . . 16 |- ( w e. ( ZZ>= ` K ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
56	55	ad2antrl 459	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
57		elfzuz3 8887	. . . . . . . . . . . . . . . 16 |- ( ( w + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
58	57	ad2antll 460	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
59		elfzuzb 8884	. . . . . . . . . . . . . . 15 |- ( ( w + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( w + 1 ) ) ) )
60	56, 58, 59	sylanbrc 394	. . . . . . . . . . . . . 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 2392	. . . . . . . . . . . . . . 15 |- ( ph -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
63	62	adantr 261	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
64		fveq2 5178	. . . . . . . . . . . . . . . 16 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
65		fveq2 5178	. . . . . . . . . . . . . . . 16 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
66	64, 65	eqeq12d 2054	. . . . . . . . . . . . . . 15 |- ( k = ( w + 1 ) -> ( ( F ` k ) = ( G ` k ) <-> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) ) )
67	66	rspcv 2652	. . . . . . . . . . . . . 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 56	. . . . . . . . . . . . 13 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
69	68	oveq2d 5528	. . . . . . . . . . . 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 2075	. . . . . . . . . . 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 2054	. . . . . . . . . 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 145	. . . . . . . . 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 357	. . . . . . . 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 23	. . . . . . 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 40	. . . . . 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 109	. . . . 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 23	. . . 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 8531	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F , S ) ` N ) = ( seq K ( .+ , G , S ) ` N ) ) ) )
79	1, 78	mpcom 32	. 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 97   = wceq 1243   e. wcel 1393   A.wral 2306   ` cfv 4902  (class class class)co 5512   1c1 6890   + caddc 6892   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-iseq 9212

This theorem is referenced by:  iseqfeq2  9229  iseqfveq  9230