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Theorem nn1suc 10343
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 9381 . . . . . 6  |-  1  e.  _V
3 nn1suc.1 . . . . . 6  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
42, 3sbcie 3221 . . . . 5  |-  ( [.
1  /  x ]. ph  <->  ps )
51, 4mpbir 209 . . . 4  |-  [. 1  /  x ]. ph
6 1nn 10333 . . . . . . 7  |-  1  e.  NN
7 eleq1 2503 . . . . . . 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 3219 . . . . . 6  |-  ( A  e.  NN  ->  ( [. A  /  x ]. ph  <->  th ) )
118, 10syl 16 . . . . 5  |-  ( A  =  1  ->  ( [. A  /  x ]. ph  <->  th ) )
12 dfsbcq 3188 . . . . 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 6116 . . . . . 6  |-  ( y  +  1 )  e. 
_V
17 nn1suc.3 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  ch ) )
1816, 17sbcie 3221 . . . . 5  |-  ( [. ( y  +  1 )  /  x ]. ph  <->  ch )
19 oveq1 6098 . . . . . 6  |-  ( y  =  ( A  - 
1 )  ->  (
y  +  1 )  =  ( ( A  -  1 )  +  1 ) )
20 dfsbcq 3188 . . . . . 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 3036 . . 3  |-  ( ( A  -  1 )  e.  NN  ->  [. (
( A  -  1 )  +  1 )  /  x ]. ph )
25 nncn 10330 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
26 ax-1cn 9340 . . . . . 6  |-  1  e.  CC
27 npcan 9619 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  +  1 )  =  A )
2825, 26, 27sylancl 662 . . . . 5  |-  ( A  e.  NN  ->  (
( A  -  1 )  +  1 )  =  A )
29 dfsbcq 3188 . . . . 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 10342 . 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 1369    e. wcel 1756   [.wsbc 3186  (class class class)co 6091   CCcc 9280   1c1 9283    + caddc 9285    - cmin 9595   NNcn 10322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-nn 10323
This theorem is referenced by:  opsqrlem6  25549
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