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Theorem cantnfp1OLD 8129
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 8103 as of 1-Jul-2019. (New usage is discouraged.) (Proof modification 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 2447 . . . . . . . 8  |-  ( Y  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( Y  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
3 eqeq1 2447 . . . . . . . 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 4893 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
74, 5, 6syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
8 eloni 4878 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
9 ordirr 4886 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
107, 8, 93syl 20 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
11 fvex 5866 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
12 dif1o 7152 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1311, 12mpbiran 918 . . . . . . . . . . . . 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 8118 . . . . . . . . . . . . . . . . . 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 5721 . . . . . . . . . . . . . . . 16  |-  ( G : B --> A  ->  G  Fn  B )
21 elpreima 5992 . . . . . . . . . . . . . . . 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 3488 . . . . . . . . . . . . . . 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 2661 . . . . . . . . . . 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 5860 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
33 simpllr 760 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3430, 32, 333eqtr4rd 2495 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
35 eqidd 2444 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
362, 3, 34, 35ifbothda 3961 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
3736mpteq2dva 4523 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
381, 37syl5eq 2496 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
3919feqmptd 5911 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4039adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4138, 40eqtr4d 2487 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4214adantr 465 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4341, 42eqeltrd 2531 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
44 oecl 7189 . . . . . . . 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 8096 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
4746, 14ffvelrnd 6017 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
48 onelon 4893 . . . . . . 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 7190 . . . . 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 6289 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
54 oecl 7189 . . . . . . . 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 7169 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
5755, 56syl 16 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
5853, 57sylan9eqr 2506 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
5958oveq1d 6296 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6041fveq2d 5860 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6152, 59, 603eqtr4rd 2495 . . 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 8126 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
71 onelon 4893 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7216, 67, 71syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  On )
73 on0eln0 4923 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7472, 73syl 16 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
7574biimpar 485 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
76 eqid 2443 . . . 4  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
77 eqid 2443 . . . 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 2443 . . . 4  |- OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
79 eqid 2443 . . . 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 8128 . . 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 2761 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 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    \ cdif 3458    C_ wss 3461   (/)c0 3770   ifcif 3926    |-> cmpt 4495    _E cep 4779   Ord word 4867   Oncon0 4868   `'ccnv 4988   dom cdm 4989   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283  seq𝜔cseqom 7114   1oc1o 7125    +o coa 7129    .o comu 7130    ^o coe 7131   Fincfn 7518  OrdIsocoi 7937   CNF ccnf 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-cnf 8082
This theorem is referenced by:  cantnflem1dOLD  8133  cantnflem1OLD  8134  cantnflem3OLD  8135
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