MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqomeq12 Structured version   Unicode version

Theorem seqomeq12 6711
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 6087 . . . . . . 7  |-  ( A  =  B  ->  (
a A b )  =  ( a B b ) )
21opeq2d 3991 . . . . . 6  |-  ( A  =  B  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
323ad2ant1 978 . . . . 5  |-  ( ( A  =  B  /\  a  e.  om  /\  b  e.  _V )  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
43mpt2eq3dva 6138 . . . 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 5728 . . . . 5  |-  ( C  =  D  ->  (  _I  `  C )  =  (  _I  `  D
) )
65opeq2d 3991 . . . 4  |-  ( C  =  D  ->  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )
7 rdgeq12 6671 . . . 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 464 . . 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 5202 . 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 6705 . 2  |- seq𝜔 ( A ,  C
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a A b ) >. ) ,  <. (/)
,  (  _I  `  C ) >. ) " om )
11 df-seqom 6705 . 2  |- seq𝜔 ( B ,  D
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. ) " om )
129, 10, 113eqtr4g 2493 1  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   <.cop 3817    _I cid 4493   suc csuc 4583   omcom 4845   "cima 4881   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   reccrdg 6667  seq𝜔cseqom 6704
This theorem is referenced by:  cantnffval  7618  cantnfval  7623  cantnfres  7633  cnfcomlem  7656  cnfcom2  7659  fin23lem33  8225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seqom 6705
  Copyright terms: Public domain W3C validator