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Theorem ee7.2aOLD 29894
Description: Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as  A mod  B. Here, just one subtraction step is proved to preserve the  gcdOLD. The  rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ee7.2aOLD  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  gcdOLD ( A ,  B )  =  gcdOLD ( A ,  ( B  -  A ) ) ) )

Proof of Theorem ee7.2aOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nndivsub 29890 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  x  e.  NN )  /\  ( ( A  /  x )  e.  NN  /\  A  <  B ) )  ->  ( ( B  /  x )  e.  NN  <->  ( ( B  -  A )  /  x )  e.  NN ) )
21exp32 605 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  x  e.  NN )  ->  (
( A  /  x
)  e.  NN  ->  ( A  <  B  -> 
( ( B  /  x )  e.  NN  <->  ( ( B  -  A
)  /  x )  e.  NN ) ) ) )
32com23 78 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  x  e.  NN )  ->  ( A  <  B  ->  (
( A  /  x
)  e.  NN  ->  ( ( B  /  x
)  e.  NN  <->  ( ( B  -  A )  /  x )  e.  NN ) ) ) )
433expia 1197 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( x  e.  NN  ->  ( A  <  B  ->  ( ( A  /  x )  e.  NN  ->  ( ( B  /  x )  e.  NN  <->  ( ( B  -  A
)  /  x )  e.  NN ) ) ) ) )
54com23 78 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  ( x  e.  NN  ->  ( ( A  /  x )  e.  NN  ->  ( ( B  /  x )  e.  NN  <->  ( ( B  -  A
)  /  x )  e.  NN ) ) ) ) )
65imp 429 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B
)  ->  ( x  e.  NN  ->  ( ( A  /  x )  e.  NN  ->  ( ( B  /  x )  e.  NN  <->  ( ( B  -  A )  /  x )  e.  NN ) ) ) )
76imp 429 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B )  /\  x  e.  NN )  ->  (
( A  /  x
)  e.  NN  ->  ( ( B  /  x
)  e.  NN  <->  ( ( B  -  A )  /  x )  e.  NN ) ) )
87pm5.32d 639 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B )  /\  x  e.  NN )  ->  (
( ( A  /  x )  e.  NN  /\  ( B  /  x
)  e.  NN )  <-> 
( ( A  /  x )  e.  NN  /\  ( ( B  -  A )  /  x
)  e.  NN ) ) )
98rabbidva 3084 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B
)  ->  { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( B  /  x )  e.  NN ) }  =  { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( ( B  -  A )  /  x )  e.  NN ) } )
109supeq1d 7904 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B
)  ->  sup ( { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( B  /  x )  e.  NN ) } ,  NN ,  <  )  =  sup ( { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  (
( B  -  A
)  /  x )  e.  NN ) } ,  NN ,  <  ) )
11 df-gcdOLD 29893 . . 3  |-  gcdOLD ( A ,  B )  =  sup ( { x  e.  NN  | 
( ( A  /  x )  e.  NN  /\  ( B  /  x
)  e.  NN ) } ,  NN ,  <  )
12 df-gcdOLD 29893 . . 3  |-  gcdOLD ( A ,  ( B  -  A ) )  =  sup ( { x  e.  NN  | 
( ( A  /  x )  e.  NN  /\  ( ( B  -  A )  /  x
)  e.  NN ) } ,  NN ,  <  )
1310, 11, 123eqtr4g 2507 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B
)  ->  gcdOLD ( A ,  B )  =  gcdOLD ( A ,  ( B  -  A ) ) )
1413ex 434 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  gcdOLD ( A ,  B )  =  gcdOLD ( A ,  ( B  -  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   {crab 2795   class class class wbr 4433  (class class class)co 6277   supcsup 7898    < clt 9626    - cmin 9805    / cdiv 10207   NNcn 10537   gcdOLDcgcdOLD 29892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-gcdOLD 29893
This theorem is referenced by: (None)
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