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Theorem seqomeq12 6482
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 5880 . . . . . . 7  |-  ( A  =  B  ->  (
a A b )  =  ( a B b ) )
21opeq2d 3819 . . . . . 6  |-  ( A  =  B  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
323ad2ant1 976 . . . . 5  |-  ( ( A  =  B  /\  a  e.  om  /\  b  e.  _V )  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
43mpt2eq3dva 5928 . . . 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 5541 . . . . 5  |-  ( C  =  D  ->  (  _I  `  C )  =  (  _I  `  D
) )
65opeq2d 3819 . . . 4  |-  ( C  =  D  ->  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )
7 rdgeq12 6442 . . . 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 463 . . 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 5027 . 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 6476 . 2  |- seq𝜔 ( A ,  C
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a A b ) >. ) ,  <. (/)
,  (  _I  `  C ) >. ) " om )
11 df-seqom 6476 . 2  |- seq𝜔 ( B ,  D
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. ) " om )
129, 10, 113eqtr4g 2353 1  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656    _I cid 4320   suc csuc 4410   omcom 4672   "cima 4708   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   reccrdg 6438  seq𝜔cseqom 6475
This theorem is referenced by:  cantnffval  7380  cantnfval  7385  cantnfres  7395  cnfcomlem  7418  cnfcom2  7421  fin23lem33  7987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seqom 6476
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