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Theorem nn1suc 9767
Description: If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. 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 8833 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 3025 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 200 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 9757 . . . . . . 7  |-  1  e.  NN
7 eleq1 2343 . . . . . . 7  |-  ( A  =  1  ->  ( A  e.  NN  <->  1  e.  NN ) )
86, 7mpbiri 224 . . . . . 6  |-  ( A  =  1  ->  A  e.  NN )
9 nn1suc.4 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  th ) )
109sbcieg 3023 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 15 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 2993 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
1311, 12bitr3d 246 . . . 4  |-  ( A  =  1  ->  ( th 
<-> 
[. 1  /  x ]. ph ) )
145, 13mpbiri 224 . . 3  |-  ( A  =  1  ->  th )
1514a1i 10 . 2  |-  ( A  e.  NN  ->  ( A  =  1  ->  th ) )
16 ovex 5883 . . . . . 6  |-  ( y  +  1 )  e. 
_V
17 nn1suc.3 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
1816, 17sbcie 3025 . . . . 5  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  ch )
19 oveq1 5865 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
20 dfsbcq 2993 . . . . . 6  |-  ( ( y  +  1 )  =  ( ( A  -  1 )  +  1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2119, 20syl 15 . . . . 5  |-  ( y  =  ( A  - 
1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2218, 21syl5bbr 250 . . . 4  |-  ( y  =  ( A  - 
1 )  ->  ( ch 
<-> 
[. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
23 nn1suc.6 . . . 4  |-  ( y  e.  NN  ->  ch )
2422, 23vtoclga 2849 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
25 nncn 9754 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
26 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
27 npcan 9060 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
2825, 26, 27sylancl 643 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
29 dfsbcq 2993 . . . . 5  |-  ( ( ( A  -  1 )  +  1 )  =  A  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3028, 29syl 15 . . . 4  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3130, 10bitrd 244 . . 3  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  th ) )
3224, 31syl5ib 210 . 2  |-  ( A  e.  NN  ->  (
( A  -  1 )  e.  NN  ->  th ) )
33 nn1m1nn 9766 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
3415, 32, 33mpjaod 370 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   [.wsbc 2991  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746
This theorem is referenced by:  opsqrlem6  22725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-nn 9747
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