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Theorem cantnfp1OLD 8122
Description: If  F is created by adding a single term  ( F `
 X )  =  Y to  G, where  X is larger than any element of the support of  G, then  F is also a finitely supported function and it is assigned the value  ( ( A  ^o  X )  .o  Y
)  +o  z where  z is the value of  G. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnfp1 8096 as of 1-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
cantnfp1OLD.4  |-  ( ph  ->  G  e.  S )
cantnfp1OLD.5  |-  ( ph  ->  X  e.  B )
cantnfp1OLD.6  |-  ( ph  ->  Y  e.  A )
cantnfp1OLD.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1OLD.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1OLD  |-  ( ph  ->  ( F  e.  S  /\  ( ( A CNF  B
) `  F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  ( ( A CNF  B
) `  G )
) ) )
Distinct variable groups:    t, B    t, A    t, S    t, G    ph, t    t, Y   
t, X
Allowed substitution hint:    F( t)

Proof of Theorem cantnfp1OLD
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfp1OLD.f . . . . . 6  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
2 eqeq1 2471 . . . . . . . 8  |-  ( Y  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( Y  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
3 eqeq1 2471 . . . . . . . 8  |-  ( ( G `  t )  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( ( G `  t )  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
4 cantnfsOLD.3 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  On )
5 cantnfp1OLD.5 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  B )
6 onelon 4903 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
74, 5, 6syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
8 eloni 4888 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
9 ordirr 4896 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
107, 8, 93syl 20 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
11 fvex 5874 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
12 dif1o 7147 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1311, 12mpbiran 916 . . . . . . . . . . . . 13  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
14 cantnfp1OLD.4 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G  e.  S )
15 cantnfsOLD.1 . . . . . . . . . . . . . . . . . . 19  |-  S  =  dom  ( A CNF  B
)
16 cantnfsOLD.2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A  e.  On )
1715, 16, 4cantnfsOLD 8111 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
1814, 17mpbid 210 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
1918simpld 459 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : B --> A )
20 ffn 5729 . . . . . . . . . . . . . . . 16  |-  ( G : B --> A  ->  G  Fn  B )
21 elpreima 5999 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2219, 20, 213syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( G `  X
)  e.  ( _V 
\  1o ) ) ) )
23 cantnfp1OLD.7 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
2423sseld 3503 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  ->  X  e.  X
) )
2522, 24sylbird 235 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X  e.  B  /\  ( G `
 X )  e.  ( _V  \  1o ) )  ->  X  e.  X ) )
265, 25mpand 675 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G `  X )  e.  ( _V  \  1o )  ->  X  e.  X
) )
2713, 26syl5bir 218 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  ->  X  e.  X ) )
2827necon1bd 2685 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  X  e.  X  ->  ( G `  X )  =  (/) ) )
2910, 28mpd 15 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  (/) )
3029ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  X )  =  (/) )
31 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  t  =  X )
3231fveq2d 5868 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
33 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3430, 32, 333eqtr4rd 2519 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
35 eqidd 2468 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
362, 3, 34, 35ifbothda 3974 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
3736mpteq2dva 4533 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
381, 37syl5eq 2520 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
3919feqmptd 5918 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4039adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4138, 40eqtr4d 2511 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4214adantr 465 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4341, 42eqeltrd 2555 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
44 oecl 7184 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
4516, 4, 44syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
4615, 16, 4cantnff 8089 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
4746, 14ffvelrnd 6020 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
48 onelon 4903 . . . . . . 7  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )  ->  (
( A CNF  B ) `
 G )  e.  On )
4945, 47, 48syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  On )
5049adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  G
)  e.  On )
51 oa0r 7185 . . . . 5  |-  ( ( ( A CNF  B ) `
 G )  e.  On  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
5250, 51syl 16 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
53 oveq2 6290 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
54 oecl 7184 . . . . . . . 8  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
5516, 7, 54syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
56 om0 7164 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
5755, 56syl 16 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
5853, 57sylan9eqr 2530 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
5958oveq1d 6297 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6041fveq2d 5868 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6152, 59, 603eqtr4rd 2519 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
6243, 61jca 532 . 2  |-  ( (
ph  /\  Y  =  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
6316adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  A  e.  On )
644adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  B  e.  On )
6514adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  G  e.  S )
665adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  X  e.  B )
67 cantnfp1OLD.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
6867adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  Y  e.  A )
6923adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( `' G " ( _V  \  1o ) )  C_  X
)
7015, 63, 64, 65, 66, 68, 69, 1cantnfp1lem1OLD 8119 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
71 onelon 4903 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7216, 67, 71syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  On )
73 on0eln0 4933 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7472, 73syl 16 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
7574biimpar 485 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
76 eqid 2467 . . . 4  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
77 eqid 2467 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
78 eqid 2467 . . . 4  |- OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
79 eqid 2467 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) ) `  k ) )  .o  ( G `  (OrdIso (  _E  ,  ( `' G " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) ) `  k ) )  .o  ( G `  (OrdIso (  _E  ,  ( `' G " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
8015, 63, 64, 65, 66, 68, 69, 1, 75, 76, 77, 78, 79cantnfp1lem3OLD 8121 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
8170, 80jca 532 . 2  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
8262, 81pm2.61dane 2785 1  |-  ( ph  ->  ( F  e.  S  /\  ( ( A CNF  B
) `  F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  ( ( A CNF  B
) `  G )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ifcif 3939    |-> cmpt 4505    _E cep 4789   Ord word 4877   Oncon0 4878   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284  seq𝜔cseqom 7109   1oc1o 7120    +o coa 7124    .o comu 7125    ^o coe 7126   Fincfn 7513  OrdIsocoi 7930   CNF ccnf 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-oexp 7133  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-cnf 8075
This theorem is referenced by:  cantnflem1dOLD  8126  cantnflem1OLD  8127  cantnflem3OLD  8128
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