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Theorem nn1suc 8699
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 7725 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 2913 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 145 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 8691 . . . . . . 7  |-  1  e.  NN
7 eleq1 2178 . . . . . . 7  |-  ( A  =  1  ->  ( A  e.  NN  <->  1  e.  NN ) )
86, 7mpbiri 167 . . . . . 6  |-  ( A  =  1  ->  A  e.  NN )
9 nn1suc.4 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  th ) )
109sbcieg 2911 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 14 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 2882 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
1311, 12bitr3d 189 . . . 4  |-  ( A  =  1  ->  ( th 
<-> 
[. 1  /  x ]. ph ) )
145, 13mpbiri 167 . . 3  |-  ( A  =  1  ->  th )
1514a1i 9 . 2  |-  ( A  e.  NN  ->  ( A  =  1  ->  th ) )
16 elisset 2672 . . . 4  |-  ( ( A  -  1 )  e.  NN  ->  E. y 
y  =  ( A  -  1 ) )
17 eleq1 2178 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1817pm5.32ri 448 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  <-> 
( ( A  - 
1 )  e.  NN  /\  y  =  ( A  -  1 ) ) )
19 nn1suc.6 . . . . . . 7  |-  ( y  e.  NN  ->  ch )
2019adantr 272 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ch )
21 nnre 8687 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  RR )
22 peano2re 7862 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  +  1 )  e.  RR )
23 nn1suc.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
2423sbcieg 2911 . . . . . . . . 9  |-  ( ( y  +  1 )  e.  RR  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2521, 22, 243syl 17 . . . . . . . 8  |-  ( y  e.  NN  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2625adantr 272 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  ch )
)
27 oveq1 5747 . . . . . . . . 9  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
2827sbceq1d 2885 . . . . . . . 8  |-  ( y  =  ( A  - 
1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2928adantl 273 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3026, 29bitr3d 189 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( ch  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3120, 30mpbid 146 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3218, 31sylbir 134 . . . 4  |-  ( ( ( A  -  1 )  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3316, 32exlimddv 1852 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
34 nncn 8688 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
35 ax-1cn 7677 . . . . . 6  |-  1  e.  CC
36 npcan 7935 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
3734, 35, 36sylancl 407 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
3837sbceq1d 2885 . . . 4  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3938, 10bitrd 187 . . 3  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  th ) )
4033, 39syl5ib 153 . 2  |-  ( A  e.  NN  ->  (
( A  -  1 )  e.  NN  ->  th ) )
41 nn1m1nn 8698 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
4215, 40, 41mpjaod 690 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   [.wsbc 2880  (class class class)co 5740   CCcc 7582   RRcr 7583   1c1 7585    + caddc 7587    - cmin 7897   NNcn 8680
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-sub 7899  df-inn 8681
This theorem is referenced by: (None)
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