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Theorem nn1suc 8125
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 7176 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 2849 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 144 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 8117 . . . . . . 7  |-  1  e.  NN
7 eleq1 2142 . . . . . . 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 2847 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 14 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 2818 . . . . 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 2614 . . . 4  |-  ( ( A  -  1 )  e.  NN  ->  E. y 
y  =  ( A  -  1 ) )
17 eleq1 2142 . . . . . 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 8113 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  RR )
22 peano2re 7311 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  +  1 )  e.  RR )
23 nn1suc.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
2423sbcieg 2847 . . . . . . . . 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 5550 . . . . . . . . 9  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
2827sbceq1d 2821 . . . . . . . 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 1820 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
34 nncn 8114 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
35 ax-1cn 7131 . . . . . 6  |-  1  e.  CC
36 npcan 7384 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
3734, 35, 36sylancl 404 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
3837sbceq1d 2821 . . . 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 8124 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
4215, 40, 41mpjaod 671 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   [.wsbc 2816  (class class class)co 5543   CCcc 7041   RRcr 7042   1c1 7044    + caddc 7046    - cmin 7346   NNcn 8106
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-inn 8107
This theorem is referenced by: (None)
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