Theorem seqeq3 12806

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

Assertion

Ref	Expression
seqeq3	|- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Proof of Theorem seqeq3

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

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . . 7 |- ( F = G -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
2	1	oveq2d 6666	. . . . . 6 |- ( F = G -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
3	2	opeq2d 4409	. . . . 5 |- ( F = G -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. )
4	3	mpt2eq3dv 6721	. . . 4 |- ( F = G -> ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) )
5		fveq1 6190	. . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
6	5	opeq2d 4409	. . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
7		rdgeq12 7509	. . . 4 |- ( ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) /\ <. M , ( F ` M ) >. = <. M , ( G ` M ) >. ) -> 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
8	4, 6, 7	syl2anc 693	. . 3 |- ( F = G -> 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
9	8	imaeq1d 5465	. 2 |- ( F = G -> ( 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 .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) " om ) )
10		df-seq 12802	. 2 |- seq M ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
11		df-seq 12802	. 2 |- seq M ( .+ , G ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) " om )
12	9, 10, 11	3eqtr4g 2681	1 |- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )

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:  seqeq3d  12809  cbvprod  14645  iprodmul  14734  geolim3  24094  leibpilem2  24668  basel  24816  faclim  31632  ovoliunnfl  33451  voliunnfl  33453  heiborlem10  33619  binomcxplemnn0  38548  binomcxplemdvsum  38554  binomcxp  38556  fourierdlem112  40435  fouriersw  40448  voliunsge0lem  40689