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Theorem nninfall 13568
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 6376 . . . . 5  |-  1o  =/=  (/)
21nesymi 2373 . . . 4  |-  -.  (/)  =  1o
3 simplr 520 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  e. )
4 nninff 7061 . . . . . . . . . . . 12  |-  ( p  e.  ->  p : om --> 2o )
54ffnd 5319 . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  Q  e.  ( 2o  ^m ) )
9 nninfall.inf . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Q `  (
x  e.  om  |->  1o ) )  =  1o )
109ad2antrr 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 13567 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  A. n  e.  om  ( p `  n
)  =  1o )
15 eqeq1 2164 . . . . . . . . . . . . . . 15  |-  ( a  =  ( p `  n )  ->  (
a  =  1o  <->  ( p `  n )  =  1o ) )
1615ralrn 5604 . . . . . . . . . . . . . 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 4552 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
20 elex2 2728 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  E. b 
b  e.  om )
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15  |-  E. b 
b  e.  om
22 fdm 5324 . . . . . . . . . . . . . . . . 17  |-  ( p : om --> 2o  ->  dom  p  =  om )
2322eleq2d 2227 . . . . . . . . . . . . . . . 16  |-  ( p : om --> 2o  ->  ( b  e.  dom  p  <->  b  e.  om ) )
2423exbidv 1805 . . . . . . . . . . . . . . 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 4804 . . . . . . . . . . . . . . 15  |-  ( E. b  b  e.  dom  p 
<->  E. a  a  e. 
ran  p )
27 eqsnm 3718 . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . 11  |-  ( ran  p  =  { 1o }  ->  ran  p  C_  { 1o } )
3331, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ran  p  C_  { 1o } )
34 df-f 5173 . . . . . . . . . 10  |-  ( p : om --> { 1o } 
<->  ( p  Fn  om  /\ 
ran  p  C_  { 1o } ) )
356, 33, 34sylanbrc 414 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p : om --> { 1o } )
36 1onn 6464 . . . . . . . . . 10  |-  1o  e.  om
37 fconst2g 5681 . . . . . . . . . 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 4632 . . . . . . . 8  |-  ( om 
X.  { 1o }
)  =  ( x  e.  om  |->  1o )
4139, 40eqtrdi 2206 . . . . . . 7  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  p  =  ( x  e.  om  |->  1o ) )
4241fveq2d 5471 . . . . . 6  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  ( Q `  p )  =  ( Q `  ( x  e.  om  |->  1o ) ) )
4342, 13, 103eqtr3d 2198 . . . . 5  |-  ( ( ( ph  /\  p  e. )  /\  ( Q `  p )  =  (/) )  ->  (/)  =  1o )
4443ex 114 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  -> 
(/)  =  1o ) )
452, 44mtoi 654 . . 3  |-  ( (
ph  /\  p  e. )  ->  -.  ( Q `  p
)  =  (/) )
46 elmapi 6612 . . . . . . 7  |-  ( Q  e.  ( 2o  ^m )  ->  Q : --> 2o )
477, 46syl 14 . . . . . 6  |-  ( ph  ->  Q : --> 2o )
4847ffvelrnda 5601 . . . . 5  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  e.  2o )
49 elpri 3583 . . . . . 6  |-  ( ( Q `  p )  e.  { (/) ,  1o }  ->  ( ( Q `
 p )  =  (/)  \/  ( Q `  p )  =  1o ) )
50 df2o3 6374 . . . . . 6  |-  2o  =  { (/) ,  1o }
5149, 50eleq2s 2252 . . . . 5  |-  ( ( Q `  p )  e.  2o  ->  (
( Q `  p
)  =  (/)  \/  ( Q `  p )  =  1o ) )
5248, 51syl 14 . . . 4  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  (/)  \/  ( Q `  p
)  =  1o ) )
5352orcomd 719 . . 3  |-  ( (
ph  /\  p  e. )  -> 
( ( Q `  p )  =  1o  \/  ( Q `  p )  =  (/) ) )
5445, 53ecased 1331 . 2  |-  ( (
ph  /\  p  e. )  -> 
( Q `  p
)  =  1o )
5554ralrimiva 2530 1  |-  ( ph  ->  A. p  e.  ( Q `  p
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435    C_ wss 3102   (/)c0 3394   ifcif 3505   {csn 3560   {cpr 3561    |-> cmpt 4025   omcom 4548    X. cxp 4583   dom cdm 4585   ran crn 4586    Fn wfn 5164   -->wf 5165   ` cfv 5169  (class class class)co 5821   1oc1o 6353   2oc2o 6354    ^m cmap 6590  ℕxnninf 7058
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1o 6360  df-2o 6361  df-map 6592  df-nninf 7059
This theorem is referenced by:  nninfsel  13576  nninffeq  13579
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