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Theorem 2submod 12016
Description: If a real number is between a positive real number and the double of the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
Assertion
Ref Expression
2submod  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )

Proof of Theorem 2submod
StepHypRef Expression
1 rpre 11226 . . . . . . 7  |-  ( B  e.  RR+  ->  B  e.  RR )
2 ax-1rid 9562 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
31, 2syl 16 . . . . . 6  |-  ( B  e.  RR+  ->  ( B  x.  1 )  =  B )
43adantl 466 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  1 )  =  B )
54oveq2d 6300 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  ( B  x.  1 ) )  =  ( A  -  B ) )
65oveq1d 6299 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( B  x.  1
) )  mod  B
)  =  ( ( A  -  B )  mod  B ) )
76adantr 465 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  ( B  x.  1 ) )  mod 
B )  =  ( ( A  -  B
)  mod  B )
)
8 simpl 457 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  RR )
9 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR+ )
10 1zzd 10895 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
1  e.  ZZ )
118, 9, 103jca 1176 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  e.  RR  /\  B  e.  RR+  /\  1  e.  ZZ ) )
1211adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  e.  RR  /\  B  e.  RR+  /\  1  e.  ZZ ) )
13 modcyc2 12000 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  1  e.  ZZ )  ->  (
( A  -  ( B  x.  1 ) )  mod  B )  =  ( A  mod  B ) )
1412, 13syl 16 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  ( B  x.  1 ) )  mod 
B )  =  ( A  mod  B ) )
15 resubcl 9883 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
161, 15sylan2 474 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  e.  RR )
1716, 9jca 532 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  B )  e.  RR  /\  B  e.  RR+ )
)
1817adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  B )  e.  RR  /\  B  e.  RR+ ) )
19 subge0 10065 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  -  B )  <->  B  <_  A ) )
201, 19sylan2 474 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( 0  <_  ( A  -  B )  <->  B  <_  A ) )
2120bicomd 201 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  <_  A  <->  0  <_  ( A  -  B ) ) )
22 rpcn 11228 . . . . . . . . 9  |-  ( B  e.  RR+  ->  B  e.  CC )
23222timesd 10781 . . . . . . . 8  |-  ( B  e.  RR+  ->  ( 2  x.  B )  =  ( B  +  B
) )
2423adantl 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( 2  x.  B
)  =  ( B  +  B ) )
2524breq2d 4459 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  <  (
2  x.  B )  <-> 
A  <  ( B  +  B ) ) )
261adantl 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
278, 26, 26ltsubaddd 10148 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  B )  <  B  <->  A  <  ( B  +  B ) ) )
2825, 27bitr4d 256 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  <  (
2  x.  B )  <-> 
( A  -  B
)  <  B )
)
2921, 28anbi12d 710 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  <_  A  /\  A  <  (
2  x.  B ) )  <->  ( 0  <_ 
( A  -  B
)  /\  ( A  -  B )  <  B
) ) )
3029biimpa 484 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( 0  <_  ( A  -  B )  /\  ( A  -  B )  <  B ) )
31 modid 11988 . . 3  |-  ( ( ( ( A  -  B )  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  ( A  -  B )  /\  ( A  -  B
)  <  B )
)  ->  ( ( A  -  B )  mod  B )  =  ( A  -  B ) )
3218, 30, 31syl2anc 661 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  B )  mod  B )  =  ( A  -  B ) )
337, 14, 323eqtr3d 2516 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447  (class class class)co 6284   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   2c2 10585   ZZcz 10864   RR+crp 11220    mod cmo 11964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fl 11897  df-mod 11965
This theorem is referenced by:  modifeq2int  12017  modaddmodup  12018
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