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Theorem ee7.2aOLD 25927
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  gcd OLD. 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  ->  gcd OLD ( A ,  B )  =  gcd OLD ( A ,  ( B  -  A ) ) ) )

Proof of Theorem ee7.2aOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nndivsub 25923 . . . . . . . . . . . 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 589 . . . . . . . . . . 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 74 . . . . . . . . . 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 1155 . . . . . . . . 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 74 . . . . . . . 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 419 . . . . . . 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 419 . . . . . 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 621 . . . . 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 2892 . . . 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 7388 . . 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 25926 . . 3  |-  gcd OLD ( A ,  B )  =  sup ( { x  e.  NN  | 
( ( A  /  x )  e.  NN  /\  ( B  /  x
)  e.  NN ) } ,  NN ,  <  )
12 df-gcdOLD 25926 . . 3  |-  gcd OLD ( A ,  ( B  -  A ) )  =  sup ( { x  e.  NN  | 
( ( A  /  x )  e.  NN  /\  ( ( B  -  A )  /  x
)  e.  NN ) } ,  NN ,  <  )
1310, 11, 123eqtr4g 2446 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  A  <  B
)  ->  gcd OLD ( A ,  B )  =  gcd OLD ( A ,  ( B  -  A ) ) )
1413ex 424 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  gcd OLD ( A ,  B )  =  gcd OLD ( A ,  ( B  -  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2655   class class class wbr 4155  (class class class)co 6022   supcsup 7382    < clt 9055    - cmin 9225    / cdiv 9611   NNcn 9934   gcd OLDcgcdOLD 25925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-gcdOLD 25926
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