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

Theorem quoremnn0ALT 12020
Description: Alternate proof of quoremnn0 12019 not using quoremz 12018. TODO - Keep either quoremnn0ALT 12020 (if we don't keep quoremz 12018) or quoremnn0 12019 (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 11992 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  NN0 )
31, 2syl5eqel 2494 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  NN0 )
4 quorem.2 . . 3  |-  R  =  ( A  -  ( B  x.  Q )
)
5 nnnn0 10842 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  NN0 )
65adantl 464 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  NN0 )
76, 3nn0mulcld 10897 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  NN0 )
8 simpl 455 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  NN0 )
93nn0cnd 10894 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  CC )
10 nncn 10583 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
1110adantl 464 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  CC )
12 nnne0 10608 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
1312adantl 464 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  =/=  0 )
149, 11, 13divcan3d 10365 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
15 nn0nndivcl 10903 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
16 flle 11971 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
1715, 16syl 17 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
181, 17syl5eqbr 4427 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
1914, 18eqbrtrd 4414 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
207nn0red 10893 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  RR )
21 nn0re 10844 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
2221adantr 463 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  RR )
23 nnre 10582 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
2423adantl 464 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  RR )
25 nngt0 10604 . . . . . . 7  |-  ( B  e.  NN  ->  0  <  B )
2625adantl 464 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  0  <  B )
27 lediv1 10447 . . . . . 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 1234 . . . . 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 10964 . . . 4  |-  ( ( ( B  x.  Q
)  e.  NN0  /\  A  e.  NN0  /\  ( B  x.  Q )  <_  A )  ->  ( A  -  ( B  x.  Q ) )  e. 
NN0 )
317, 8, 29, 30syl3anc 1230 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
324, 31syl5eqel 2494 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  NN0 )
331oveq2i 6288 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
34 fraclt1 11974 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
3515, 34syl 17 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
3633, 35syl5eqbr 4427 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
374oveq1i 6287 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
38 nn0cn 10845 . . . . . . . . 9  |-  ( A  e.  NN0  ->  A  e.  CC )
3938adantr 463 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  CC )
407nn0cnd 10894 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  CC )
4110, 12jca 530 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
4241adantl 464 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
43 divsubdir 10280 . . . . . . . 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 1230 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
4514oveq2d 6293 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
4644, 45eqtrd 2443 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
4737, 46syl5eq 2455 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
4810, 12dividd 10358 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
4948adantl 464 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
5036, 47, 493brtr4d 4424 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  <  ( B  /  B ) )
5132nn0red 10893 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  RR )
52 ltdiv1 10446 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
5351, 24, 24, 26, 52syl112anc 1234 . . . 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 6288 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
5640, 39pncan3d 9969 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
5755, 56syl5req 2456 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
5854, 57jca 530 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
593, 32, 58jca31 532 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    < clt 9657    <_ cle 9658    - cmin 9840    / cdiv 10246   NNcn 10575   NN0cn0 10835   |_cfl 11962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-fl 11964
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator