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Theorem nn1suc 8940
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 7954 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 2999 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 146 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 8932 . . . . . . 7  |-  1  e.  NN
7 eleq1 2240 . . . . . . 7  |-  ( A  =  1  ->  ( A  e.  NN  <->  1  e.  NN ) )
86, 7mpbiri 168 . . . . . 6  |-  ( A  =  1  ->  A  e.  NN )
9 nn1suc.4 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  th ) )
109sbcieg 2997 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 14 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 2966 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
1311, 12bitr3d 190 . . . 4  |-  ( A  =  1  ->  ( th 
<-> 
[. 1  /  x ]. ph ) )
145, 13mpbiri 168 . . 3  |-  ( A  =  1  ->  th )
1514a1i 9 . 2  |-  ( A  e.  NN  ->  ( A  =  1  ->  th ) )
16 elisset 2753 . . . 4  |-  ( ( A  -  1 )  e.  NN  ->  E. y 
y  =  ( A  -  1 ) )
17 eleq1 2240 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1817pm5.32ri 455 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  <-> 
( ( A  - 
1 )  e.  NN  /\  y  =  ( A  -  1 ) ) )
19 nn1suc.6 . . . . . . 7  |-  ( y  e.  NN  ->  ch )
2019adantr 276 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ch )
21 nnre 8928 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  RR )
22 peano2re 8095 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  +  1 )  e.  RR )
23 nn1suc.3 . . . . . . . . . 10  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
2423sbcieg 2997 . . . . . . . . 9  |-  ( ( y  +  1 )  e.  RR  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2521, 22, 243syl 17 . . . . . . . 8  |-  ( y  e.  NN  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  ch ) )
2625adantr 276 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  ch )
)
27 oveq1 5884 . . . . . . . . 9  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
2827sbceq1d 2969 . . . . . . . 8  |-  ( y  =  ( A  - 
1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2928adantl 277 . . . . . . 7  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( [. (
y  +  1 )  /  x ]. ph  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3026, 29bitr3d 190 . . . . . 6  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  ( ch  <->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
3120, 30mpbid 147 . . . . 5  |-  ( ( y  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3218, 31sylbir 135 . . . 4  |-  ( ( ( A  -  1 )  e.  NN  /\  y  =  ( A  -  1 ) )  ->  [. ( ( A  -  1 )  +  1 )  /  x ]. ph )
3316, 32exlimddv 1898 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
34 nncn 8929 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
35 ax-1cn 7906 . . . . . 6  |-  1  e.  CC
36 npcan 8168 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
3734, 35, 36sylancl 413 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
3837sbceq1d 2969 . . . 4  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3938, 10bitrd 188 . . 3  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  th ) )
4033, 39imbitrid 154 . 2  |-  ( A  e.  NN  ->  (
( A  -  1 )  e.  NN  ->  th ) )
41 nn1m1nn 8939 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
4215, 40, 41mpjaod 718 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   [.wsbc 2964  (class class class)co 5877   CCcc 7811   RRcr 7812   1c1 7814    + caddc 7816    - cmin 8130   NNcn 8921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-inn 8922
This theorem is referenced by: (None)
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