Theorem seqeq2 12093

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

Assertion

Ref	Expression
seqeq2	|- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )

Proof of Theorem seqeq2

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

Step	Hyp	Ref	Expression
1		oveq 6276	. . . . . 6 |- ( .+ = Q -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y Q ( F ` ( x + 1 ) ) ) )
2	1	opeq2d 4210	. . . . 5 |- ( .+ = Q -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. )
3	2	mpt2eq3dv 6336	. . . 4 |- ( .+ = Q -> ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) )
4		rdgeq1 7069	. . . 4 |- ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) -> 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 Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
5	3, 4	syl 16	. . 3 |- ( .+ = Q -> 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 Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
6	5	imaeq1d 5324	. 2 |- ( .+ = Q -> ( 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 Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om ) )
7		df-seq 12090	. 2 |- seq M ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
8		df-seq 12090	. 2 |- seq M ( Q , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
9	6, 7, 8	3eqtr4g 2520	1 |- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 3106   <.cop 4022   "cima 4991   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   omcom 6673   reccrdg 7067   1c1 9482   + caddc 9484   seqcseq 12089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seq 12090

This theorem is referenced by:  seqeq2d  12096  sadcom  14197  gxfval  25457  ressmulgnn  27905  cvmliftlem15  29007