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Theorem ltmod 31886
Description: A sufficient condition for a "less than" relationship for the  mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ltmod.a  |-  ( ph  ->  A  e.  RR )
ltmod.b  |-  ( ph  ->  B  e.  RR+ )
ltmod.c  |-  ( ph  ->  C  e.  ( ( A  -  ( A  mod  B ) ) [,) A ) )
Assertion
Ref Expression
ltmod  |-  ( ph  ->  ( C  mod  B
)  <  ( A  mod  B ) )

Proof of Theorem ltmod
StepHypRef Expression
1 ltmod.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
2 ltmod.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR+ )
31, 2modcld 11984 . . . . . . 7  |-  ( ph  ->  ( A  mod  B
)  e.  RR )
41, 3resubcld 9983 . . . . . 6  |-  ( ph  ->  ( A  -  ( A  mod  B ) )  e.  RR )
51rexrd 9632 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
6 icossre 11608 . . . . . 6  |-  ( ( ( A  -  ( A  mod  B ) )  e.  RR  /\  A  e.  RR* )  ->  (
( A  -  ( A  mod  B ) ) [,) A )  C_  RR )
74, 5, 6syl2anc 659 . . . . 5  |-  ( ph  ->  ( ( A  -  ( A  mod  B ) ) [,) A ) 
C_  RR )
8 ltmod.c . . . . 5  |-  ( ph  ->  C  e.  ( ( A  -  ( A  mod  B ) ) [,) A ) )
97, 8sseldd 3490 . . . 4  |-  ( ph  ->  C  e.  RR )
102rpred 11259 . . . . 5  |-  ( ph  ->  B  e.  RR )
119, 2rerpdivcld 11286 . . . . . . 7  |-  ( ph  ->  ( C  /  B
)  e.  RR )
1211flcld 11916 . . . . . 6  |-  ( ph  ->  ( |_ `  ( C  /  B ) )  e.  ZZ )
1312zred 10965 . . . . 5  |-  ( ph  ->  ( |_ `  ( C  /  B ) )  e.  RR )
1410, 13remulcld 9613 . . . 4  |-  ( ph  ->  ( B  x.  ( |_ `  ( C  /  B ) ) )  e.  RR )
154rexrd 9632 . . . . 5  |-  ( ph  ->  ( A  -  ( A  mod  B ) )  e.  RR* )
16 icoltub 31787 . . . . 5  |-  ( ( ( A  -  ( A  mod  B ) )  e.  RR*  /\  A  e. 
RR*  /\  C  e.  ( ( A  -  ( A  mod  B ) ) [,) A ) )  ->  C  <  A )
1715, 5, 8, 16syl3anc 1226 . . . 4  |-  ( ph  ->  C  <  A )
189, 1, 14, 17ltsub1dd 10160 . . 3  |-  ( ph  ->  ( C  -  ( B  x.  ( |_ `  ( C  /  B
) ) ) )  <  ( A  -  ( B  x.  ( |_ `  ( C  /  B ) ) ) ) )
19 icossicc 11614 . . . . . . . 8  |-  ( ( A  -  ( A  mod  B ) ) [,) A )  C_  ( ( A  -  ( A  mod  B ) ) [,] A )
2019, 8sseldi 3487 . . . . . . 7  |-  ( ph  ->  C  e.  ( ( A  -  ( A  mod  B ) ) [,] A ) )
211, 2, 20lefldiveq 31725 . . . . . 6  |-  ( ph  ->  ( |_ `  ( A  /  B ) )  =  ( |_ `  ( C  /  B
) ) )
2221eqcomd 2462 . . . . 5  |-  ( ph  ->  ( |_ `  ( C  /  B ) )  =  ( |_ `  ( A  /  B
) ) )
2322oveq2d 6286 . . . 4  |-  ( ph  ->  ( B  x.  ( |_ `  ( C  /  B ) ) )  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) )
2423oveq2d 6286 . . 3  |-  ( ph  ->  ( A  -  ( B  x.  ( |_ `  ( C  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
2518, 24breqtrd 4463 . 2  |-  ( ph  ->  ( C  -  ( B  x.  ( |_ `  ( C  /  B
) ) ) )  <  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
26 modval 11980 . . 3  |-  ( ( C  e.  RR  /\  B  e.  RR+ )  -> 
( C  mod  B
)  =  ( C  -  ( B  x.  ( |_ `  ( C  /  B ) ) ) ) )
279, 2, 26syl2anc 659 . 2  |-  ( ph  ->  ( C  mod  B
)  =  ( C  -  ( B  x.  ( |_ `  ( C  /  B ) ) ) ) )
28 modval 11980 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
291, 2, 28syl2anc 659 . 2  |-  ( ph  ->  ( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
3025, 27, 293brtr4d 4469 1  |-  ( ph  ->  ( C  mod  B
)  <  ( A  mod  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480    x. cmul 9486   RR*cxr 9616    < clt 9617    - cmin 9796    / cdiv 10202   RR+crp 11221   [,)cico 11534   [,]cicc 11535   |_cfl 11908    mod cmo 11978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fl 11910  df-mod 11979
This theorem is referenced by:  fouriersw  32256
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