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Theorem nninfall 11369
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 6153 . . . . 5  |-  1o  =/=  (/)
21nesymi 2297 . . . 4  |-  -.  (/)  =  1o
3 simplr 497 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  e. )
4 nninff 11363 . . . . . . . . . . . 12  |-  ( p  e.  ->  p : om --> 2o )
54ffnd 5129 . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  Q  e.  ( 2o  ^m ) )
9 nninfall.inf . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Q `  (
x  e.  om  |->  1o ) )  =  1o )
109ad2antrr 472 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )  =  1o )
13 simpr 108 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  (/) )
148, 10, 12, 3, 13nninfalllem1 11368 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( p `  n
)  =  1o )
15 eqeq1 2091 . . . . . . . . . . . . . . 15  |-  ( a  =  ( p `  n )  ->  (
a  =  1o  <->  ( p `  n )  =  1o ) )
1615ralrn 5402 . . . . . . . . . . . . . 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 165 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. a  e.  ran  p  a  =  1o )
19 peano1 4384 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
20 elex2 2629 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  E. b 
b  e.  om )
2119, 20ax-mp 7 . . . . . . . . . . . . . . 15  |-  E. b 
b  e.  om
22 fdm 5133 . . . . . . . . . . . . . . . . 17  |-  ( p : om --> 2o  ->  dom  p  =  om )
2322eleq2d 2154 . . . . . . . . . . . . . . . 16  |-  ( p : om --> 2o  ->  ( b  e.  dom  p  <->  b  e.  om ) )
2423exbidv 1750 . . . . . . . . . . . . . . 15  |-  ( p : om --> 2o  ->  ( E. b  b  e. 
dom  p  <->  E. b 
b  e.  om )
)
2521, 24mpbiri 166 . . . . . . . . . . . . . 14  |-  ( p : om --> 2o  ->  E. b  b  e.  dom  p )
26 dmmrnm 4625 . . . . . . . . . . . . . . 15  |-  ( E. b  b  e.  dom  p 
<->  E. a  a  e. 
ran  p )
27 eqsnm 3584 . . . . . . . . . . . . . . 15  |-  ( E. a  a  e.  ran  p  ->  ( ran  p  =  { 1o }  <->  A. a  e.  ran  p  a  =  1o ) )
2826, 27sylbi 119 . . . . . . . . . . . . . 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 165 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  =  { 1o } )
32 eqimss 3067 . . . . . . . . . . 11  |-  ( ran  p  =  { 1o }  ->  ran  p  C_  { 1o } )
3331, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  C_  { 1o } )
34 df-f 4987 . . . . . . . . . 10  |-  ( p : om --> { 1o } 
<->  ( p  Fn  om  /\ 
ran  p  C_  { 1o } ) )
356, 33, 34sylanbrc 408 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p : om --> { 1o } )
36 1onn 6233 . . . . . . . . . 10  |-  1o  e.  om
37 fconst2g 5475 . . . . . . . . . 10  |-  ( 1o  e.  om  ->  (
p : om --> { 1o } 
<->  p  =  ( om 
X.  { 1o }
) ) )
3836, 37ax-mp 7 . . . . . . . . 9  |-  ( p : om --> { 1o } 
<->  p  =  ( om 
X.  { 1o }
) )
3935, 38sylib 120 . . . . . . . 8  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( om  X.  { 1o } ) )
40 fconstmpt 4455 . . . . . . . 8  |-  ( om 
X.  { 1o }
)  =  ( x  e.  om  |->  1o )
4139, 40syl6eq 2133 . . . . . . 7  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( x  e.  om  |->  1o ) )
4241fveq2d 5274 . . . . . 6  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  ( Q `  ( x  e.  om  |->  1o ) ) )
4342, 13, 103eqtr3d 2125 . . . . 5  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  (/)  =  1o )
4443ex 113 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  -> 
(/)  =  1o ) )
452, 44mtoi 623 . . 3  |-  ( (
ph  /\  p  e. )  ->  -.  ( Q `  p
)  =  (/) )
46 elmapi 6381 . . . . . . 7  |-  ( Q  e.  ( 2o  ^m )  ->  Q : --> 2o )
477, 46syl 14 . . . . . 6  |-  ( ph  ->  Q : --> 2o )
4847ffvelrnda 5399 . . . . 5  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  e.  2o )
49 elpri 3454 . . . . . 6  |-  ( ( Q `  p )  e.  { (/) ,  1o }  ->  ( ( Q `
 p )  =  (/)  \/  ( Q `  p )  =  1o ) )
50 df2o3 6151 . . . . . 6  |-  2o  =  { (/) ,  1o }
5149, 50eleq2s 2179 . . . . 5  |-  ( ( Q `  p )  e.  2o  ->  (
( Q `  p
)  =  (/)  \/  ( Q `  p )  =  1o ) )
5248, 51syl 14 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  \/  ( Q `  p
)  =  1o ) )
5352orcomd 681 . . 3  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  1o  \/  ( Q `  p )  =  (/) ) )
5445, 53ecased 1283 . 2  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  =  1o )
5554ralrimiva 2442 1  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287   E.wex 1424    e. wcel 1436   A.wral 2355    C_ wss 2988   (/)c0 3275   ifcif 3379   {csn 3431   {cpr 3432    |-> cmpt 3876   omcom 4380    X. cxp 4411   dom cdm 4413   ran crn 4414    Fn wfn 4978   -->wf 4979   ` cfv 4983  (class class class)co 5615   1oc1o 6130   2oc2o 6131    ^m cmap 6359  ℕxnninf 6736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1o 6137  df-2o 6138  df-map 6361  df-nninf 6738
This theorem is referenced by:  nninfsel  11378
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