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Theorem nninfall 14042
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 6411 . . . . 5  |-  1o  =/=  (/)
21nesymi 2386 . . . 4  |-  -.  (/)  =  1o
3 simplr 525 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  e. )
4 nninff 7099 . . . . . . . . . . . 12  |-  ( p  e.  ->  p : om --> 2o )
54ffnd 5348 . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  Q  e.  ( 2o  ^m ) )
9 nninfall.inf . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Q `  (
x  e.  om  |->  1o ) )  =  1o )
109ad2antrr 485 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 14041 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( p `  n
)  =  1o )
15 eqeq1 2177 . . . . . . . . . . . . . . 15  |-  ( a  =  ( p `  n )  ->  (
a  =  1o  <->  ( p `  n )  =  1o ) )
1615ralrn 5634 . . . . . . . . . . . . . 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 4578 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
20 elex2 2746 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  E. b 
b  e.  om )
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15  |-  E. b 
b  e.  om
22 fdm 5353 . . . . . . . . . . . . . . . . 17  |-  ( p : om --> 2o  ->  dom  p  =  om )
2322eleq2d 2240 . . . . . . . . . . . . . . . 16  |-  ( p : om --> 2o  ->  ( b  e.  dom  p  <->  b  e.  om ) )
2423exbidv 1818 . . . . . . . . . . . . . . 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 4830 . . . . . . . . . . . . . . 15  |-  ( E. b  b  e.  dom  p 
<->  E. a  a  e. 
ran  p )
27 eqsnm 3742 . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . 11  |-  ( ran  p  =  { 1o }  ->  ran  p  C_  { 1o } )
3331, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  C_  { 1o } )
34 df-f 5202 . . . . . . . . . 10  |-  ( p : om --> { 1o } 
<->  ( p  Fn  om  /\ 
ran  p  C_  { 1o } ) )
356, 33, 34sylanbrc 415 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p : om --> { 1o } )
36 1onn 6499 . . . . . . . . . 10  |-  1o  e.  om
37 fconst2g 5711 . . . . . . . . . 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 4658 . . . . . . . 8  |-  ( om 
X.  { 1o }
)  =  ( x  e.  om  |->  1o )
4139, 40eqtrdi 2219 . . . . . . 7  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( x  e.  om  |->  1o ) )
4241fveq2d 5500 . . . . . 6  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  ( Q `  ( x  e.  om  |->  1o ) ) )
4342, 13, 103eqtr3d 2211 . . . . 5  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  (/)  =  1o )
4443ex 114 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  -> 
(/)  =  1o ) )
452, 44mtoi 659 . . 3  |-  ( (
ph  /\  p  e. )  ->  -.  ( Q `  p
)  =  (/) )
46 elmapi 6648 . . . . . . 7  |-  ( Q  e.  ( 2o  ^m )  ->  Q : --> 2o )
477, 46syl 14 . . . . . 6  |-  ( ph  ->  Q : --> 2o )
4847ffvelrnda 5631 . . . . 5  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  e.  2o )
49 elpri 3606 . . . . . 6  |-  ( ( Q `  p )  e.  { (/) ,  1o }  ->  ( ( Q `
 p )  =  (/)  \/  ( Q `  p )  =  1o ) )
50 df2o3 6409 . . . . . 6  |-  2o  =  { (/) ,  1o }
5149, 50eleq2s 2265 . . . . 5  |-  ( ( Q `  p )  e.  2o  ->  (
( Q `  p
)  =  (/)  \/  ( Q `  p )  =  1o ) )
5248, 51syl 14 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  \/  ( Q `  p
)  =  1o ) )
5352orcomd 724 . . 3  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  1o  \/  ( Q `  p )  =  (/) ) )
5445, 53ecased 1344 . 2  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  =  1o )
5554ralrimiva 2543 1  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    = wceq 1348   E.wex 1485    e. wcel 2141   A.wral 2448    C_ wss 3121   (/)c0 3414   ifcif 3526   {csn 3583   {cpr 3584    |-> cmpt 4050   omcom 4574    X. cxp 4609   dom cdm 4611   ran crn 4612    Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853   1oc1o 6388   2oc2o 6389    ^m cmap 6626  ℕxnninf 7096
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-nninf 7097
This theorem is referenced by:  nninfsel  14050  nninffeq  14053
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