MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfp1 Structured version   Unicode version

Theorem cantnfp1 7993
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 4845 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
52, 3, 4syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
6 eloni 4830 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
7 ordirr 4838 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
85, 6, 73syl 20 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
9 fvex 5802 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
10 dif1o 7043 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
119, 10mpbiran 909 . . . . . . . . . . . . 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 7978 . . . . . . . . . . . . . . . . . . . 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 5660 . . . . . . . . . . . . . . . . . 18  |-  ( G : B --> A  ->  G  Fn  B )
1917, 18syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  Fn  B )
20 0ex 4523 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  _V
2120a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  -> 
(/)  e.  _V )
22 elsuppfn 6801 . . . . . . . . . . . . . . . . 17  |-  ( ( G  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2319, 2, 21, 22syl3anc 1219 . . . . . . . . . . . . . . . 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 3456 . . . . . . . . . . . . . . 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 2666 . . . . . . . . . . 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 5796 . . . . . . . . 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 2503 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
40 eqidd 2452 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
4139, 40ifeqda 3923 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
4241mpteq2dva 4479 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
431, 42syl5eq 2504 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
4417feqmptd 5846 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4544adantr 465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4643, 45eqtr4d 2495 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4712adantr 465 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4846, 47eqeltrd 2539 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
49 oecl 7080 . . . . . . . 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 7986 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
5251, 12ffvelrnd 5946 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
53 onelon 4845 . . . . . . 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 7081 . . . . 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 6201 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
59 oecl 7080 . . . . . . . 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 7060 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
6260, 61syl 16 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
6358, 62sylan9eqr 2514 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
6463oveq1d 6208 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6546fveq2d 5796 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6657, 64, 653eqtr4rd 2503 . . 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 7990 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
76 onelon 4845 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7714, 72, 76syl2anc 661 . . . . . 6  |-  ( ph  ->  Y  e.  On )
78 on0eln0 4875 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7977, 78syl 16 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
8079biimpar 485 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
81 eqid 2451 . . . 4  |- OrdIso (  _E  ,  ( F supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) )
82 eqid 2451 . . . 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 2451 . . . 4  |- OrdIso (  _E  ,  ( G supp  (/) ) )  = OrdIso (  _E  , 
( G supp  (/) ) )
84 eqid 2451 . . . 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 7992 . . 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 2766 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 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071    \ cdif 3426    C_ wss 3429   (/)c0 3738   ifcif 3892   class class class wbr 4393    |-> cmpt 4451    _E cep 4731   Ord word 4819   Oncon0 4820   dom cdm 4941    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   supp csupp 6793  seq𝜔cseqom 7005   1oc1o 7016    +o coa 7020    .o comu 7021    ^o coe 7022   finSupp cfsupp 7724  OrdIsocoi 7827   CNF ccnf 7971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-seqom 7006  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-oexp 7029  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-cnf 7972
This theorem is referenced by:  cantnflem1d  8000  cantnflem1  8001  cantnflem3  8003
  Copyright terms: Public domain W3C validator