Theorem seqeq1 12804

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( M = N -> ( F ` M ) = ( F ` N ) )
2		opeq12 4404	. . . . 5 |- ( ( M = N /\ ( F ` M ) = ( F ` N ) ) -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
3	1, 2	mpdan 702	. . . 4 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		rdgeq2 7508	. . . 4 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 17	. . 3 |- ( M = N -> rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6	5	imaeq1d 5465	. 2 |- ( M = N -> ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) " om ) )
7		df-seq 12802	. 2 |- seq M ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
8		df-seq 12802	. 2 |- seq N ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) " om )
9	6, 7, 8	3eqtr4g 2681	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   <.cop 4183   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   reccrdg 7505   1c1 9937   + caddc 9939   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802

This theorem is referenced by:  seqeq1d  12807  seqfn  12813  seq1  12814  seqp1  12816  seqf1olem2  12841  seqid  12846  seqz  12849  iserex  14387  summolem2  14447  summo  14448  zsum  14449  isumsplit  14572  ntrivcvg  14629  ntrivcvgn0  14630  ntrivcvgtail  14632  ntrivcvgmullem  14633  prodmolem2  14665  prodmo  14666  zprod  14667  fprodntriv  14672  ege2le3  14820  gsumval2a  17279  leibpi  24669  dvradcnv2  38546  binomcxplemnotnn0  38555  stirlinglem12  40302