Theorem seq1st 15284

Description: A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
algrf.1	|- Z = ( ZZ>= ` M )
algrf.2	|- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )

Assertion

Ref	Expression
seq1st	|- ( ( M e. ZZ /\ A e. V ) -> R = seq M ( ( F o. 1st ) , { <. M , A >. } ) )

Proof of Theorem seq1st

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algrf.2	. 2 |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )
2		seqfn 12813	. . . 4 |- ( M e. ZZ -> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) Fn ( ZZ>= ` M ) )
3	2	adantr 481	. . 3 |- ( ( M e. ZZ /\ A e. V ) -> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) Fn ( ZZ>= ` M ) )
4		seqfn 12813	. . . 4 |- ( M e. ZZ -> seq M ( ( F o. 1st ) , { <. M , A >. } ) Fn ( ZZ>= ` M ) )
5	4	adantr 481	. . 3 |- ( ( M e. ZZ /\ A e. V ) -> seq M ( ( F o. 1st ) , { <. M , A >. } ) Fn ( ZZ>= ` M ) )
6		fveq2 6191	. . . . . . . 8 |- ( y = M -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) )
7		fveq2 6191	. . . . . . . 8 |- ( y = M -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) )
8	6, 7	eqeq12d 2637	. . . . . . 7 |- ( y = M -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) <-> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) ) )
9	8	imbi2d 330	. . . . . 6 |- ( y = M -> ( ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) ) <-> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) ) ) )
10		fveq2 6191	. . . . . . . 8 |- ( y = x -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) )
11		fveq2 6191	. . . . . . . 8 |- ( y = x -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) )
12	10, 11	eqeq12d 2637	. . . . . . 7 |- ( y = x -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) <-> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) )
13	12	imbi2d 330	. . . . . 6 |- ( y = x -> ( ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) ) <-> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) ) )
14		fveq2 6191	. . . . . . . 8 |- ( y = ( x + 1 ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) )
15		fveq2 6191	. . . . . . . 8 |- ( y = ( x + 1 ) -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) )
16	14, 15	eqeq12d 2637	. . . . . . 7 |- ( y = ( x + 1 ) -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) <-> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) ) )
17	16	imbi2d 330	. . . . . 6 |- ( y = ( x + 1 ) -> ( ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` y ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` y ) ) <-> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) ) ) )
18		seq1 12814	. . . . . . . . 9 |- ( M e. ZZ -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( ( Z X. { A } ) ` M ) )
19	18	adantr 481	. . . . . . . 8 |- ( ( M e. ZZ /\ A e. V ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( ( Z X. { A } ) ` M ) )
20		seq1 12814	. . . . . . . . . 10 |- ( M e. ZZ -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) = ( { <. M , A >. } ` M ) )
21	20	adantr 481	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. V ) -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) = ( { <. M , A >. } ` M ) )
22		id 22	. . . . . . . . . . 11 |- ( A e. V -> A e. V )
23		uzid 11702	. . . . . . . . . . . 12 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
24		algrf.1	. . . . . . . . . . . 12 |- Z = ( ZZ>= ` M )
25	23, 24	syl6eleqr 2712	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. Z )
26		fvconst2g 6467	. . . . . . . . . . 11 |- ( ( A e. V /\ M e. Z ) -> ( ( Z X. { A } ) ` M ) = A )
27	22, 25, 26	syl2anr 495	. . . . . . . . . 10 |- ( ( M e. ZZ /\ A e. V ) -> ( ( Z X. { A } ) ` M ) = A )
28		fvsng 6447	. . . . . . . . . 10 |- ( ( M e. ZZ /\ A e. V ) -> ( { <. M , A >. } ` M ) = A )
29	27, 28	eqtr4d 2659	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. V ) -> ( ( Z X. { A } ) ` M ) = ( { <. M , A >. } ` M ) )
30	21, 29	eqtr4d 2659	. . . . . . . 8 |- ( ( M e. ZZ /\ A e. V ) -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) = ( ( Z X. { A } ) ` M ) )
31	19, 30	eqtr4d 2659	. . . . . . 7 |- ( ( M e. ZZ /\ A e. V ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) )
32	31	ex 450	. . . . . 6 |- ( M e. ZZ -> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` M ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` M ) ) )
33		fveq2 6191	. . . . . . . . 9 |- ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) -> ( F ` ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ) = ( F ` ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) )
34		seqp1 12816	. . . . . . . . . . . 12 |- ( x e. ( ZZ>= ` M ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ( F o. 1st ) ( ( Z X. { A } ) ` ( x + 1 ) ) ) )
35		fvex 6201	. . . . . . . . . . . . 13 |- ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) e. _V
36		fvex 6201	. . . . . . . . . . . . 13 |- ( ( Z X. { A } ) ` ( x + 1 ) ) e. _V
37	35, 36	algrflem 7286	. . . . . . . . . . . 12 |- ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ( F o. 1st ) ( ( Z X. { A } ) ` ( x + 1 ) ) ) = ( F ` ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) )
38	34, 37	syl6eq 2672	. . . . . . . . . . 11 |- ( x e. ( ZZ>= ` M ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( F ` ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ) )
39		seqp1 12816	. . . . . . . . . . . 12 |- ( x e. ( ZZ>= ` M ) -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) = ( ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ( F o. 1st ) ( { <. M , A >. } ` ( x + 1 ) ) ) )
40		fvex 6201	. . . . . . . . . . . . 13 |- ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) e. _V
41		fvex 6201	. . . . . . . . . . . . 13 |- ( { <. M , A >. } ` ( x + 1 ) ) e. _V
42	40, 41	algrflem 7286	. . . . . . . . . . . 12 |- ( ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ( F o. 1st ) ( { <. M , A >. } ` ( x + 1 ) ) ) = ( F ` ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) )
43	39, 42	syl6eq 2672	. . . . . . . . . . 11 |- ( x e. ( ZZ>= ` M ) -> ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) = ( F ` ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) )
44	38, 43	eqeq12d 2637	. . . . . . . . . 10 |- ( x e. ( ZZ>= ` M ) -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) <-> ( F ` ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ) = ( F ` ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) ) )
45	44	adantl 482	. . . . . . . . 9 |- ( ( A e. V /\ x e. ( ZZ>= ` M ) ) -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) <-> ( F ` ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) ) = ( F ` ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) ) )
46	33, 45	syl5ibr 236	. . . . . . . 8 |- ( ( A e. V /\ x e. ( ZZ>= ` M ) ) -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) ) )
47	46	expcom 451	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> ( A e. V -> ( ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) ) ) )
48	47	a2d 29	. . . . . 6 |- ( x e. ( ZZ>= ` M ) -> ( ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) -> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` ( x + 1 ) ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` ( x + 1 ) ) ) ) )
49	9, 13, 17, 13, 32, 48	uzind4 11746	. . . . 5 |- ( x e. ( ZZ>= ` M ) -> ( A e. V -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) ) )
50	49	impcom 446	. . . 4 |- ( ( A e. V /\ x e. ( ZZ>= ` M ) ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) )
51	50	adantll 750	. . 3 |- ( ( ( M e. ZZ /\ A e. V ) /\ x e. ( ZZ>= ` M ) ) -> ( seq M ( ( F o. 1st ) , ( Z X. { A } ) ) ` x ) = ( seq M ( ( F o. 1st ) , { <. M , A >. } ) ` x ) )
52	3, 5, 51	eqfnfvd 6314	. 2 |- ( ( M e. ZZ /\ A e. V ) -> seq M ( ( F o. 1st ) , ( Z X. { A } ) ) = seq M ( ( F o. 1st ) , { <. M , A >. } ) )
53	1, 52	syl5eq 2668	1 |- ( ( M e. ZZ /\ A e. V ) -> R = seq M ( ( F o. 1st ) , { <. M , A >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   X. cxp 5112   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1stc1st 7166   1c1 9937   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802

This theorem is referenced by: (None)