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Theorem nninfall 13290
Description: Given a decidable predicate on ℕ, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which  Q is a decidable predicate is that it assigns a value of either  (/) or  1o (which can be thought of as false and true) to every element of ℕ. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
Hypotheses
Ref Expression
nninfall.q  |-  ( ph  ->  Q  e.  ( 2o 
^m ) )
nninfall.inf  |-  ( ph  ->  ( Q `  (
x  e.  om  |->  1o ) )  =  1o )
nninfall.n  |-  ( ph  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )  =  1o )
Assertion
Ref Expression
nninfall  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Distinct variable groups:    Q, n, i   
n, p, i, ph
Allowed substitution hints:    ph( x)    Q( x, p)

Proof of Theorem nninfall
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 6329 . . . . 5  |-  1o  =/=  (/)
21nesymi 2354 . . . 4  |-  -.  (/)  =  1o
3 simplr 519 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  e. )
4 nninff 13284 . . . . . . . . . . . 12  |-  ( p  e.  ->  p : om --> 2o )
54ffnd 5273 . . . . . . . . . . 11  |-  ( p  e.  ->  p  Fn  om )
63, 5syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  Fn  om )
7 nninfall.q . . . . . . . . . . . . . . 15  |-  ( ph  ->  Q  e.  ( 2o 
^m ) )
87ad2antrr 479 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  Q  e.  ( 2o  ^m ) )
9 nninfall.inf . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Q `  (
x  e.  om  |->  1o ) )  =  1o )
109ad2antrr 479 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  ( x  e.  om  |->  1o ) )  =  1o )
11 nninfall.n . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )  =  1o )
1211ad2antrr 479 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )  =  1o )
13 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  (/) )
148, 10, 12, 3, 13nninfalllem1 13289 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( p `  n
)  =  1o )
15 eqeq1 2146 . . . . . . . . . . . . . . 15  |-  ( a  =  ( p `  n )  ->  (
a  =  1o  <->  ( p `  n )  =  1o ) )
1615ralrn 5558 . . . . . . . . . . . . . 14  |-  ( p  Fn  om  ->  ( A. a  e.  ran  p  a  =  1o  <->  A. n  e.  om  (
p `  n )  =  1o ) )
173, 5, 163syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( A. a  e.  ran  p  a  =  1o  <->  A. n  e.  om  ( p `  n
)  =  1o ) )
1814, 17mpbird 166 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. a  e.  ran  p  a  =  1o )
19 peano1 4508 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
20 elex2 2702 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  E. b 
b  e.  om )
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15  |-  E. b 
b  e.  om
22 fdm 5278 . . . . . . . . . . . . . . . . 17  |-  ( p : om --> 2o  ->  dom  p  =  om )
2322eleq2d 2209 . . . . . . . . . . . . . . . 16  |-  ( p : om --> 2o  ->  ( b  e.  dom  p  <->  b  e.  om ) )
2423exbidv 1797 . . . . . . . . . . . . . . 15  |-  ( p : om --> 2o  ->  ( E. b  b  e. 
dom  p  <->  E. b 
b  e.  om )
)
2521, 24mpbiri 167 . . . . . . . . . . . . . 14  |-  ( p : om --> 2o  ->  E. b  b  e.  dom  p )
26 dmmrnm 4758 . . . . . . . . . . . . . . 15  |-  ( E. b  b  e.  dom  p 
<->  E. a  a  e. 
ran  p )
27 eqsnm 3682 . . . . . . . . . . . . . . 15  |-  ( E. a  a  e.  ran  p  ->  ( ran  p  =  { 1o }  <->  A. a  e.  ran  p  a  =  1o ) )
2826, 27sylbi 120 . . . . . . . . . . . . . 14  |-  ( E. b  b  e.  dom  p  ->  ( ran  p  =  { 1o }  <->  A. a  e.  ran  p  a  =  1o ) )
2925, 28syl 14 . . . . . . . . . . . . 13  |-  ( p : om --> 2o  ->  ( ran  p  =  { 1o }  <->  A. a  e.  ran  p  a  =  1o ) )
303, 4, 293syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( ran  p  =  { 1o }  <->  A. a  e.  ran  p  a  =  1o ) )
3118, 30mpbird 166 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  =  { 1o } )
32 eqimss 3151 . . . . . . . . . . 11  |-  ( ran  p  =  { 1o }  ->  ran  p  C_  { 1o } )
3331, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  C_  { 1o } )
34 df-f 5127 . . . . . . . . . 10  |-  ( p : om --> { 1o } 
<->  ( p  Fn  om  /\ 
ran  p  C_  { 1o } ) )
356, 33, 34sylanbrc 413 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p : om --> { 1o } )
36 1onn 6416 . . . . . . . . . 10  |-  1o  e.  om
37 fconst2g 5635 . . . . . . . . . 10  |-  ( 1o  e.  om  ->  (
p : om --> { 1o } 
<->  p  =  ( om 
X.  { 1o }
) ) )
3836, 37ax-mp 5 . . . . . . . . 9  |-  ( p : om --> { 1o } 
<->  p  =  ( om 
X.  { 1o }
) )
3935, 38sylib 121 . . . . . . . 8  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( om  X.  { 1o } ) )
40 fconstmpt 4586 . . . . . . . 8  |-  ( om 
X.  { 1o }
)  =  ( x  e.  om  |->  1o )
4139, 40syl6eq 2188 . . . . . . 7  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( x  e.  om  |->  1o ) )
4241fveq2d 5425 . . . . . 6  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  ( Q `  ( x  e.  om  |->  1o ) ) )
4342, 13, 103eqtr3d 2180 . . . . 5  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  (/)  =  1o )
4443ex 114 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  -> 
(/)  =  1o ) )
452, 44mtoi 653 . . 3  |-  ( (
ph  /\  p  e. )  ->  -.  ( Q `  p
)  =  (/) )
46 elmapi 6564 . . . . . . 7  |-  ( Q  e.  ( 2o  ^m )  ->  Q : --> 2o )
477, 46syl 14 . . . . . 6  |-  ( ph  ->  Q : --> 2o )
4847ffvelrnda 5555 . . . . 5  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  e.  2o )
49 elpri 3550 . . . . . 6  |-  ( ( Q `  p )  e.  { (/) ,  1o }  ->  ( ( Q `
 p )  =  (/)  \/  ( Q `  p )  =  1o ) )
50 df2o3 6327 . . . . . 6  |-  2o  =  { (/) ,  1o }
5149, 50eleq2s 2234 . . . . 5  |-  ( ( Q `  p )  e.  2o  ->  (
( Q `  p
)  =  (/)  \/  ( Q `  p )  =  1o ) )
5248, 51syl 14 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  \/  ( Q `  p
)  =  1o ) )
5352orcomd 718 . . 3  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  1o  \/  ( Q `  p )  =  (/) ) )
5445, 53ecased 1327 . 2  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  =  1o )
5554ralrimiva 2505 1  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2416    C_ wss 3071   (/)c0 3363   ifcif 3474   {csn 3527   {cpr 3528    |-> cmpt 3989   omcom 4504    X. cxp 4537   dom cdm 4539   ran crn 4540    Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   1oc1o 6306   2oc2o 6307    ^m cmap 6542  ℕxnninf 7005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-nninf 7007
This theorem is referenced by:  nninfsel  13299  nninffeq  13302
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