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Theorem nninfall 13100
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 6297 . . . . 5  |-  1o  =/=  (/)
21nesymi 2331 . . . 4  |-  -.  (/)  =  1o
3 simplr 504 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  e. )
4 nninff 13094 . . . . . . . . . . . 12  |-  ( p  e.  ->  p : om --> 2o )
54ffnd 5243 . . . . . . . . . . 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 13099 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( p `  n
)  =  1o )
15 eqeq1 2124 . . . . . . . . . . . . . . 15  |-  ( a  =  ( p `  n )  ->  (
a  =  1o  <->  ( p `  n )  =  1o ) )
1615ralrn 5526 . . . . . . . . . . . . . 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 4478 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
20 elex2 2676 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  E. b 
b  e.  om )
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15  |-  E. b 
b  e.  om
22 fdm 5248 . . . . . . . . . . . . . . . . 17  |-  ( p : om --> 2o  ->  dom  p  =  om )
2322eleq2d 2187 . . . . . . . . . . . . . . . 16  |-  ( p : om --> 2o  ->  ( b  e.  dom  p  <->  b  e.  om ) )
2423exbidv 1781 . . . . . . . . . . . . . . 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 4728 . . . . . . . . . . . . . . 15  |-  ( E. b  b  e.  dom  p 
<->  E. a  a  e. 
ran  p )
27 eqsnm 3652 . . . . . . . . . . . . . . 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 3121 . . . . . . . . . . 11  |-  ( ran  p  =  { 1o }  ->  ran  p  C_  { 1o } )
3331, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  C_  { 1o } )
34 df-f 5097 . . . . . . . . . 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 6384 . . . . . . . . . 10  |-  1o  e.  om
37 fconst2g 5603 . . . . . . . . . 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 4556 . . . . . . . 8  |-  ( om 
X.  { 1o }
)  =  ( x  e.  om  |->  1o )
4139, 40syl6eq 2166 . . . . . . 7  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( x  e.  om  |->  1o ) )
4241fveq2d 5393 . . . . . 6  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  ( Q `  ( x  e.  om  |->  1o ) ) )
4342, 13, 103eqtr3d 2158 . . . . 5  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  (/)  =  1o )
4443ex 114 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  -> 
(/)  =  1o ) )
452, 44mtoi 638 . . 3  |-  ( (
ph  /\  p  e. )  ->  -.  ( Q `  p
)  =  (/) )
46 elmapi 6532 . . . . . . 7  |-  ( Q  e.  ( 2o  ^m )  ->  Q : --> 2o )
477, 46syl 14 . . . . . 6  |-  ( ph  ->  Q : --> 2o )
4847ffvelrnda 5523 . . . . 5  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  e.  2o )
49 elpri 3520 . . . . . 6  |-  ( ( Q `  p )  e.  { (/) ,  1o }  ->  ( ( Q `
 p )  =  (/)  \/  ( Q `  p )  =  1o ) )
50 df2o3 6295 . . . . . 6  |-  2o  =  { (/) ,  1o }
5149, 50eleq2s 2212 . . . . 5  |-  ( ( Q `  p )  e.  2o  ->  (
( Q `  p
)  =  (/)  \/  ( Q `  p )  =  1o ) )
5248, 51syl 14 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  \/  ( Q `  p
)  =  1o ) )
5352orcomd 703 . . 3  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  1o  \/  ( Q `  p )  =  (/) ) )
5445, 53ecased 1312 . 2  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  =  1o )
5554ralrimiva 2482 1  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    = wceq 1316   E.wex 1453    e. wcel 1465   A.wral 2393    C_ wss 3041   (/)c0 3333   ifcif 3444   {csn 3497   {cpr 3498    |-> cmpt 3959   omcom 4474    X. cxp 4507   dom cdm 4509   ran crn 4510    Fn wfn 5088   -->wf 5089   ` cfv 5093  (class class class)co 5742   1oc1o 6274   2oc2o 6275    ^m cmap 6510  ℕxnninf 6973
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1o 6281  df-2o 6282  df-map 6512  df-nninf 6975
This theorem is referenced by:  nninfsel  13109  nninffeq  13112
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