Theorem seqeq2 12805

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 6656	. . . . . 6 |- ( .+ = Q -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y Q ( F ` ( x + 1 ) ) ) )
2	1	opeq2d 4409	. . . . 5 |- ( .+ = Q -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y Q ( F ` ( x + 1 ) ) ) >. )
3	2	mpt2eq3dv 6721	. . . 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 7507	. . . 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 17	. . 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 5465	. 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 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 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 2681	1 |- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , 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-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802

This theorem is referenced by:  seqeq2d  12808  sadcom  15185  ressmulgnn  29683  cvmliftlem15  31280