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Theorem seqomeq12 7182
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 6311 . . . . . 6  |-  ( A  =  B  ->  (
a A b )  =  ( a B b ) )
21opeq2d 4194 . . . . 5  |-  ( A  =  B  ->  <. suc  a ,  ( a A b ) >.  =  <. suc  a ,  ( a B b ) >.
)
32mpt2eq3dv 6371 . . . 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 5881 . . . . 5  |-  ( C  =  D  ->  (  _I  `  C )  =  (  _I  `  D
) )
54opeq2d 4194 . . . 4  |-  ( C  =  D  ->  <. (/) ,  (  _I  `  C )
>.  =  <. (/) ,  (  _I  `  D )
>. )
6 rdgeq12 7142 . . . 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 479 . . 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 5186 . 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 7176 . 2  |- seq𝜔 ( A ,  C
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a A b ) >. ) ,  <. (/)
,  (  _I  `  C ) >. ) " om )
10 df-seqom 7176 . 2  |- seq𝜔 ( B ,  D
)  =  ( rec ( ( a  e. 
om ,  b  e. 
_V  |->  <. suc  a , 
( a B b ) >. ) ,  <. (/)
,  (  _I  `  D ) >. ) " om )
118, 9, 103eqtr4g 2488 1  |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   _Vcvv 3080   (/)c0 3761   <.cop 4004    _I cid 4763   "cima 4856   suc csuc 5444   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   omcom 6706   reccrdg 7138  seq𝜔cseqom 7175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-xp 4859  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-iota 5565  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-seqom 7176
This theorem is referenced by:  cantnffval  8176  cantnfval  8181  cantnfres  8190  cnfcomlem  8212  cnfcom2  8215  fin23lem33  8782
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