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Theorem seqomeq12 6908
Description: Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
seqomeq12  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )

Proof of Theorem seqomeq12
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 6096 . . . . . . 7  |-  ( A  =  B  ->  (
a A b )  =  ( a B b ) )
21opeq2d 4065 . . . . . 6  |-  ( A  =  B  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
323ad2ant1 1009 . . . . 5  |-  ( ( A  =  B  /\  a  e.  om  /\  b  e.  _V )  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
43mpt2eq3dva 6149 . . . 4  |-  ( A  =  B  ->  (
a  e.  om , 
b  e.  _V  |->  <. suc  a ,  ( a A b ) >.
)  =  ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a B b ) >. )
)
5 fveq2 5690 . . . . 5  |-  ( C  =  D  ->  (  _I  `  C )  =  (  _I  `  D
) )
65opeq2d 4065 . . . 4  |-  ( C  =  D  ->  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )
7 rdgeq12 6868 . . . 4  |-  ( ( ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a A b )
>. )  =  (
a  e.  om , 
b  e.  _V  |->  <. suc  a ,  ( a B b ) >.
)  /\  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )  ->  rec (
( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a A b )
>. ) ,  <. (/) ,  (  _I  `  C )
>. )  =  rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. )
)
84, 6, 7syl2an 477 . . 3  |-  ( ( A  =  B  /\  C  =  D )  ->  rec ( ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a A b ) >. ) ,  <. (/) ,  (  _I 
`  C ) >.
)  =  rec (
( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a B b )
>. ) ,  <. (/) ,  (  _I  `  D )
>. ) )
98imaeq1d 5167 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( rec ( ( a  e.  om , 
b  e.  _V  |->  <. suc  a ,  ( a A b ) >.
) ,  <. (/) ,  (  _I  `  C )
>. ) " om )  =  ( rec (
( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a B b )
>. ) ,  <. (/) ,  (  _I  `  D )
>. ) " om )
)
10 df-seqom 6902 . 2  |- seq𝜔 ( A ,  C
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a A b ) >. ) ,  <. (/)
,  (  _I  `  C ) >. ) " om )
11 df-seqom 6902 . 2  |- seq𝜔 ( B ,  D
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. ) " om )
129, 10, 113eqtr4g 2499 1  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971   (/)c0 3636   <.cop 3882    _I cid 4630   suc csuc 4720   "cima 4842   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   omcom 6475   reccrdg 6864  seq𝜔cseqom 6901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-recs 6831  df-rdg 6865  df-seqom 6902
This theorem is referenced by:  cantnffval  7868  cantnfval  7875  cantnfres  7884  cantnfvalOLD  7905  cnfcomlem  7931  cnfcom2  7934  cnfcomlemOLD  7939  cnfcom2OLD  7942  fin23lem33  8513
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