ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn1suc Unicode version

Theorem nn1suc 8413
Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
Hypotheses
Ref Expression
nn1suc.1  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
nn1suc.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
nn1suc.4  |-  ( x  =  A  ->  ( ph 
<->  th ) )
nn1suc.5  |-  ps
nn1suc.6  |-  ( y  e.  NN  ->  ch )
Assertion
Ref Expression
nn1suc  |-  ( A  e.  NN  ->  th )
Distinct variable groups:    x, y, A    ps, x    ch, x    th, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)

Proof of Theorem nn1suc
StepHypRef Expression
1 nn1suc.5 . . . . 5  |-  ps
2 1ex 7462 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 2871 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 144 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 8405 . . . . . . 7  |-  1  e.  NN
7 eleq1 2150 . . . . . . 7  |-  ( A  =  1  ->  ( A  e.  NN  <->  1  e.  NN ) )
86, 7mpbiri 166 . . . . . 6  |-  ( A  =  1  ->  A  e.  NN )
9 nn1suc.4 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  th ) )
109sbcieg 2869 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 14 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 2840 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
1311, 12bitr3d 188 . . . 4  |-  ( A  =  1  ->  ( th 
<-> 
[. 1  /  x ]. ph ) )
145, 13mpbiri 166 . . 3  |-  ( A  =  1  ->  th )
1514a1i 9 . 2  |-  ( A  e.  NN  ->  ( A  =  1  ->  th ) )
16 elisset 2633 . . . 4  |-  ( ( A  -  1 )  e.  NN  ->  E. y 
y  =  ( A  -  1 ) )
17 eleq1 2150 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1817pm5.32ri 443 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  <-> 
( ( A  - 
1 )  e.  NN  /\  y  =  ( A  -  1 ) ) )
19 nn1suc.6 . . . . . . 7  |-  ( y  e.  NN  ->  ch )
2019adantr 270 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ch )
21 nnre 8401 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  RR )
22 peano2re 7597 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  +  1 )  e.  RR )
23 nn1suc.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
2423sbcieg 2869 . . . . . . . . 9  |-  ( ( y  +  1 )  e.  RR  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2521, 22, 243syl 17 . . . . . . . 8  |-  ( y  e.  NN  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2625adantr 270 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  ch )
)
27 oveq1 5641 . . . . . . . . 9  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
2827sbceq1d 2843 . . . . . . . 8  |-  ( y  =  ( A  - 
1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2928adantl 271 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3026, 29bitr3d 188 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( ch  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3120, 30mpbid 145 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3218, 31sylbir 133 . . . 4  |-  ( ( ( A  -  1 )  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3316, 32exlimddv 1826 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
34 nncn 8402 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
35 ax-1cn 7417 . . . . . 6  |-  1  e.  CC
36 npcan 7670 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
3734, 35, 36sylancl 404 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
3837sbceq1d 2843 . . . 4  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3938, 10bitrd 186 . . 3  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  th ) )
4033, 39syl5ib 152 . 2  |-  ( A  e.  NN  ->  (
( A  -  1 )  e.  NN  ->  th ) )
41 nn1m1nn 8412 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
4215, 40, 41mpjaod 673 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   [.wsbc 2838  (class class class)co 5634   CCcc 7327   RRcr 7328   1c1 7330    + caddc 7332    - cmin 7632   NNcn 8394
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-inn 8395
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator