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Theorem nn1suc 10339
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 9377 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 3218 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 209 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 10329 . . . . . . 7  |-  1  e.  NN
7 eleq1 2501 . . . . . . 7  |-  ( A  =  1  ->  ( A  e.  NN  <->  1  e.  NN ) )
86, 7mpbiri 233 . . . . . 6  |-  ( A  =  1  ->  A  e.  NN )
9 nn1suc.4 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  th ) )
109sbcieg 3216 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 16 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 3185 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  [. 1  /  x ]. ph ) )
1311, 12bitr3d 255 . . . 4  |-  ( A  =  1  ->  ( th 
<-> 
[. 1  /  x ]. ph ) )
145, 13mpbiri 233 . . 3  |-  ( A  =  1  ->  th )
1514a1i 11 . 2  |-  ( A  e.  NN  ->  ( A  =  1  ->  th ) )
16 ovex 6115 . . . . . 6  |-  ( y  +  1 )  e. 
_V
17 nn1suc.3 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
1816, 17sbcie 3218 . . . . 5  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  ch )
19 oveq1 6097 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
20 dfsbcq 3185 . . . . . 6  |-  ( ( y  +  1 )  =  ( ( A  -  1 )  +  1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2119, 20syl 16 . . . . 5  |-  ( y  =  ( A  - 
1 )  ->  ( [. ( y  +  1 )  /  x ]. ph  <->  [. ( ( A  - 
1 )  +  1 )  /  x ]. ph ) )
2218, 21syl5bbr 259 . . . 4  |-  ( y  =  ( A  - 
1 )  ->  ( ch 
<-> 
[. ( ( A  -  1 )  +  1 )  /  x ]. ph ) )
23 nn1suc.6 . . . 4  |-  ( y  e.  NN  ->  ch )
2422, 23vtoclga 3033 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
25 nncn 10326 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
26 ax-1cn 9336 . . . . . 6  |-  1  e.  CC
27 npcan 9615 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
2825, 26, 27sylancl 657 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
29 dfsbcq 3185 . . . . 5  |-  ( ( ( A  -  1 )  +  1 )  =  A  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3028, 29syl 16 . . . 4  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  [. A  /  x ]. ph ) )
3130, 10bitrd 253 . . 3  |-  ( A  e.  NN  ->  ( [. ( ( A  - 
1 )  +  1 )  /  x ]. ph  <->  th ) )
3224, 31syl5ib 219 . 2  |-  ( A  e.  NN  ->  (
( A  -  1 )  e.  NN  ->  th ) )
33 nn1m1nn 10338 . 2  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
3415, 32, 33mpjaod 381 1  |-  ( A  e.  NN  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1364    e. wcel 1761   [.wsbc 3183  (class class class)co 6090   CCcc 9276   1c1 9279    + caddc 9281    - cmin 9591   NNcn 10318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-nn 10319
This theorem is referenced by:  opsqrlem6  25484
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