MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quoremnn0ALT Structured version   Unicode version

Theorem quoremnn0ALT 11701
Description: Alternate proof of quoremnn0 11700 not using quoremz 11699. TODO - Keep either quoremnn0ALT 11701 (if we don't keep quoremz 11699) or quoremnn0 11700 (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
quorem.1  |-  Q  =  ( |_ `  ( A  /  B ) )
quorem.2  |-  R  =  ( A  -  ( B  x.  Q )
)
Assertion
Ref Expression
quoremnn0ALT  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )

Proof of Theorem quoremnn0ALT
StepHypRef Expression
1 quorem.1 . . 3  |-  Q  =  ( |_ `  ( A  /  B ) )
2 fldivnn0 11673 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  NN0 )
31, 2syl5eqel 2527 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  NN0 )
4 quorem.2 . . 3  |-  R  =  ( A  -  ( B  x.  Q )
)
5 nnnn0 10591 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  NN0 )
65adantl 466 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  NN0 )
76, 3nn0mulcld 10646 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  NN0 )
8 simpl 457 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  NN0 )
93nn0cnd 10643 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  CC )
10 nncn 10335 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
1110adantl 466 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  CC )
12 nnne0 10359 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
1312adantl 466 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  =/=  0 )
149, 11, 13divcan3d 10117 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
15 nn0nndivcl 10650 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
16 flle 11654 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
1715, 16syl 16 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
181, 17syl5eqbr 4330 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
1914, 18eqbrtrd 4317 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
207nn0red 10642 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  RR )
21 nn0re 10593 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
2221adantr 465 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 10334 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
2423adantl 466 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  RR )
25 nngt0 10356 . . . . . . 7  |-  ( B  e.  NN  ->  0  <  B )
2625adantl 466 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  0  <  B )
27 lediv1 10199 . . . . . 6  |-  ( ( ( B  x.  Q
)  e.  RR  /\  A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2820, 22, 24, 26, 27syl112anc 1222 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
2919, 28mpbird 232 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  <_  A )
30 nn0sub2 10710 . . . 4  |-  ( ( ( B  x.  Q
)  e.  NN0  /\  A  e.  NN0  /\  ( B  x.  Q )  <_  A )  ->  ( A  -  ( B  x.  Q ) )  e. 
NN0 )
317, 8, 29, 30syl3anc 1218 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
324, 31syl5eqel 2527 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  NN0 )
331oveq2i 6107 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
34 fraclt1 11657 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
3515, 34syl 16 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
3633, 35syl5eqbr 4330 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
374oveq1i 6106 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
38 nn0cn 10594 . . . . . . . . 9  |-  ( A  e.  NN0  ->  A  e.  CC )
3938adantr 465 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  CC )
407nn0cnd 10643 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  CC )
4110, 12jca 532 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
4241adantl 466 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
43 divsubdir 10032 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( B  x.  Q
)  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  -  ( B  x.  Q ) )  /  B )  =  ( ( A  /  B
)  -  ( ( B  x.  Q )  /  B ) ) )
4439, 40, 42, 43syl3anc 1218 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
4514oveq2d 6112 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
4644, 45eqtrd 2475 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
4737, 46syl5eq 2487 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
4810, 12dividd 10110 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
4948adantl 466 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
5036, 47, 493brtr4d 4327 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  <  ( B  /  B ) )
5132nn0red 10642 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  RR )
52 ltdiv1 10198 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5351, 24, 24, 26, 52syl112anc 1222 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5450, 53mpbird 232 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  <  B )
554oveq2i 6107 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
5640, 39pncan3d 9727 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
5755, 56syl5req 2488 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
5854, 57jca 532 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
593, 32, 58jca31 534 1  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   NNcn 10327   NN0cn0 10584   |_cfl 11645
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fl 11647
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator