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Theorem cantnfp1 8131
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 5434 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
52, 3, 4syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
6 eloni 5419 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
7 ordirr 5427 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
85, 6, 73syl 20 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
9 fvex 5858 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
10 dif1o 7186 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
119, 10mpbiran 919 . . . . . . . . . . . . 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 8116 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
1612, 15mpbid 210 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1716simpld 457 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G : B --> A )
18 ffn 5713 . . . . . . . . . . . . . . . . . 18  |-  ( G : B --> A  ->  G  Fn  B )
1917, 18syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  Fn  B )
20 0ex 4525 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  _V
2120a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  -> 
(/)  e.  _V )
22 elsuppfn 6909 . . . . . . . . . . . . . . . . 17  |-  ( ( G  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2319, 2, 21, 22syl3anc 1230 . . . . . . . . . . . . . . . 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 702 . . . . . . . . . . . . . . . 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 3440 . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G `  X )  e.  ( _V  \  1o )  ->  X  e.  X
) )
3211, 31syl5bir 218 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  ->  X  e.  X ) )
3332necon1bd 2621 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  X  e.  X  ->  ( G `  X )  =  (/) ) )
348, 33mpd 15 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  (/) )
3534ad3antrrr 728 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  X )  =  (/) )
36 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  t  =  X )
3736fveq2d 5852 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
38 simpllr 761 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3935, 37, 383eqtr4rd 2454 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
40 eqidd 2403 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
4139, 40ifeqda 3917 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
4241mpteq2dva 4480 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
431, 42syl5eq 2455 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
4417feqmptd 5901 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4544adantr 463 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4643, 45eqtr4d 2446 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4712adantr 463 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4846, 47eqeltrd 2490 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
49 oecl 7223 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
5014, 2, 49syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5113, 14, 2cantnff 8124 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
5251, 12ffvelrnd 6009 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
53 onelon 5434 . . . . . . 7  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )  ->  (
( A CNF  B ) `
 G )  e.  On )
5450, 52, 53syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  On )
5554adantr 463 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  G
)  e.  On )
56 oa0r 7224 . . . . 5  |-  ( ( ( A CNF  B ) `
 G )  e.  On  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
5755, 56syl 17 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
58 oveq2 6285 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
59 oecl 7223 . . . . . . . 8  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
6014, 5, 59syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
61 om0 7203 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
6260, 61syl 17 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
6358, 62sylan9eqr 2465 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
6463oveq1d 6292 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6546fveq2d 5852 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6657, 64, 653eqtr4rd 2454 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
6748, 66jca 530 . 2  |-  ( (
ph  /\  Y  =  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
6814adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  A  e.  On )
692adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  B  e.  On )
7012adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  G  e.  S )
713adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  X  e.  B )
72 cantnfp1.y . . . . 5  |-  ( ph  ->  Y  e.  A )
7372adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  Y  e.  A )
7428adantr 463 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( G supp  (/) )  C_  X )
7513, 68, 69, 70, 71, 73, 74, 1cantnfp1lem1 8128 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
76 onelon 5434 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7714, 72, 76syl2anc 659 . . . . . 6  |-  ( ph  ->  Y  e.  On )
78 on0eln0 5464 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7977, 78syl 17 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
8079biimpar 483 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
81 eqid 2402 . . . 4  |- OrdIso (  _E  ,  ( F supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) )
82 eqid 2402 . . . 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 2402 . . . 4  |- OrdIso (  _E  ,  ( G supp  (/) ) )  = OrdIso (  _E  , 
( G supp  (/) ) )
84 eqid 2402 . . . 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 8130 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
8675, 85jca 530 . 2  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
8767, 86pm2.61dane 2721 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058    \ cdif 3410    C_ wss 3413   (/)c0 3737   ifcif 3884   class class class wbr 4394    |-> cmpt 4452    _E cep 4731   dom cdm 4822   Ord word 5408   Oncon0 5409    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   supp csupp 6901  seq𝜔cseqom 7148   1oc1o 7159    +o coa 7163    .o comu 7164    ^o coe 7165   finSupp cfsupp 7862  OrdIsocoi 7967   CNF ccnf 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-oexp 7172  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-oi 7968  df-cnf 8110
This theorem is referenced by:  cantnflem1d  8138  cantnflem1  8139  cantnflem3  8141
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