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Theorem seqomeq12 7155
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 6283 . . . . . 6  |-  ( A  =  B  ->  (
a A b )  =  ( a B b ) )
21opeq2d 4165 . . . . 5  |-  ( A  =  B  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
32mpt2eq3dv 6343 . . . 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 ) >. )
)
4 fveq2 5848 . . . . 5  |-  ( C  =  D  ->  (  _I  `  C )  =  (  _I  `  D
) )
54opeq2d 4165 . . . 4  |-  ( C  =  D  ->  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )
6 rdgeq12 7115 . . . 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 ) >. )
)
73, 5, 6syl2an 475 . . 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 )
>. ) )
87imaeq1d 5155 . 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 )
)
9 df-seqom 7149 . 2  |- seq𝜔 ( A ,  C
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a A b ) >. ) ,  <. (/)
,  (  _I  `  C ) >. ) " om )
10 df-seqom 7149 . 2  |- seq𝜔 ( B ,  D
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. ) " om )
118, 9, 103eqtr4g 2468 1  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   _Vcvv 3058   (/)c0 3737   <.cop 3977    _I cid 4732   "cima 4825   suc csuc 5411   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   omcom 6682   reccrdg 7111  seq𝜔cseqom 7148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-iota 5532  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149
This theorem is referenced by:  cantnffval  8111  cantnfval  8118  cantnfres  8127  cantnfvalOLD  8148  cnfcomlem  8174  cnfcom2  8177  cnfcomlemOLD  8182  cnfcom2OLD  8185  fin23lem33  8756
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