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Theorem cantnfp1 8096
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.) (Revised by AV, 1-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfp1.g  |-  ( ph  ->  G  e.  S )
cantnfp1.x  |-  ( ph  ->  X  e.  B )
cantnfp1.y  |-  ( ph  ->  Y  e.  A )
cantnfp1.s  |-  ( ph  ->  ( G supp  (/) )  C_  X )
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1  |-  ( 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 cantnfp1
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfp1.f . . . . . 6  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
2 cantnfs.b . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  On )
3 cantnfp1.x . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  B )
4 onelon 4903 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
52, 3, 4syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
6 eloni 4888 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
7 ordirr 4896 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
85, 6, 73syl 20 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
9 fvex 5874 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
10 dif1o 7147 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
119, 10mpbiran 916 . . . . . . . . . . . . 13  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
12 cantnfp1.g . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  G  e.  S )
13 cantnfs.s . . . . . . . . . . . . . . . . . . . . 21  |-  S  =  dom  ( A CNF  B
)
14 cantnfs.a . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  A  e.  On )
1513, 14, 2cantnfs 8081 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
1612, 15mpbid 210 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1716simpld 459 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G : B --> A )
18 ffn 5729 . . . . . . . . . . . . . . . . . 18  |-  ( G : B --> A  ->  G  Fn  B )
1917, 18syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  Fn  B )
20 0ex 4577 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  _V
2120a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  -> 
(/)  e.  _V )
22 elsuppfn 6906 . . . . . . . . . . . . . . . . 17  |-  ( ( G  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2319, 2, 21, 22syl3anc 1228 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2411bicomi 202 . . . . . . . . . . . . . . . . . 18  |-  ( ( G `  X )  =/=  (/)  <->  ( G `  X )  e.  ( _V  \  1o ) )
2524a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  <->  ( G `  X )  e.  ( _V  \  1o ) ) )
2625anbi2d 703 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( X  e.  B  /\  ( G `
 X )  =/=  (/) )  <->  ( X  e.  B  /\  ( G `
 X )  e.  ( _V  \  1o ) ) ) )
2723, 26bitrd 253 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
28 cantnfp1.s . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G supp  (/) )  C_  X )
2928sseld 3503 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( G supp  (/) )  ->  X  e.  X ) )
3027, 29sylbird 235 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X  e.  B  /\  ( G `
 X )  e.  ( _V  \  1o ) )  ->  X  e.  X ) )
313, 30mpand 675 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G `  X )  e.  ( _V  \  1o )  ->  X  e.  X
) )
3211, 31syl5bir 218 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  ->  X  e.  X ) )
3332necon1bd 2685 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  X  e.  X  ->  ( G `  X )  =  (/) ) )
348, 33mpd 15 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  (/) )
3534ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  X )  =  (/) )
36 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  t  =  X )
3736fveq2d 5868 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
38 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3935, 37, 383eqtr4rd 2519 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
40 eqidd 2468 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
4139, 40ifeqda 3972 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
4241mpteq2dva 4533 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
431, 42syl5eq 2520 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
4417feqmptd 5918 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4544adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4643, 45eqtr4d 2511 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4712adantr 465 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4846, 47eqeltrd 2555 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
49 oecl 7184 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
5014, 2, 49syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5113, 14, 2cantnff 8089 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
5251, 12ffvelrnd 6020 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
53 onelon 4903 . . . . . . 7  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )  ->  (
( A CNF  B ) `
 G )  e.  On )
5450, 52, 53syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  On )
5554adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  G
)  e.  On )
56 oa0r 7185 . . . . 5  |-  ( ( ( A CNF  B ) `
 G )  e.  On  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
5755, 56syl 16 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
58 oveq2 6290 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
59 oecl 7184 . . . . . . . 8  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
6014, 5, 59syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
61 om0 7164 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
6260, 61syl 16 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
6358, 62sylan9eqr 2530 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
6463oveq1d 6297 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6546fveq2d 5868 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6657, 64, 653eqtr4rd 2519 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
6748, 66jca 532 . 2  |-  ( (
ph  /\  Y  =  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
6814adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  A  e.  On )
692adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  B  e.  On )
7012adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  G  e.  S )
713adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  X  e.  B )
72 cantnfp1.y . . . . 5  |-  ( ph  ->  Y  e.  A )
7372adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  Y  e.  A )
7428adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( G supp  (/) )  C_  X )
7513, 68, 69, 70, 71, 73, 74, 1cantnfp1lem1 8093 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
76 onelon 4903 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7714, 72, 76syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  On )
78 on0eln0 4933 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7977, 78syl 16 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
8079biimpar 485 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
81 eqid 2467 . . . 4  |- OrdIso (  _E  ,  ( F supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) )
82 eqid 2467 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `
 k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp 
(/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )
83 eqid 2467 . . . 4  |- OrdIso (  _E  ,  ( G supp  (/) ) )  = OrdIso (  _E  , 
( G supp  (/) ) )
84 eqid 2467 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( G supp  (/) ) ) `
 k ) )  .o  ( G `  (OrdIso (  _E  ,  ( G supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( G supp  (/) ) ) `  k ) )  .o  ( G `  (OrdIso (  _E  ,  ( G supp 
(/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )
8513, 68, 69, 70, 71, 73, 74, 1, 80, 81, 82, 83, 84cantnfp1lem3 8095 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
8675, 85jca 532 . 2  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
8767, 86pm2.61dane 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   class class class wbr 4447    |-> cmpt 4505    _E cep 4789   Ord word 4877   Oncon0 4878   dom cdm 4999    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   supp csupp 6898  seq𝜔cseqom 7109   1oc1o 7120    +o coa 7124    .o comu 7125    ^o coe 7126   finSupp cfsupp 7825  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:  cantnflem1d  8103  cantnflem1  8104  cantnflem3  8106
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