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Theorem quoremnn0ALT 10977
Description: Quotient and remainder of a nonnegative integer divided by a natural number. TO DO - Keep either quoremnn0ALT 10977 ((if we don't keep quoremz 10975) or quoremnn0 10976 (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 nn0re 9990 . . . . . 6  |-  ( A  e.  NN0  ->  A  e.  RR )
32adantr 451 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  RR )
4 nnre 9769 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  RR )
54adantl 452 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  RR )
6 nnne0 9794 . . . . . 6  |-  ( B  e.  NN  ->  B  =/=  0 )
76adantl 452 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  =/=  0 )
8 redivcl 9495 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
93, 5, 7, 8syl3anc 1182 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  /  B
)  e.  RR )
10 nn0ge0 10007 . . . . . 6  |-  ( A  e.  NN0  ->  0  <_  A )
112, 10jca 518 . . . . 5  |-  ( A  e.  NN0  ->  ( A  e.  RR  /\  0  <_  A ) )
12 nngt0 9791 . . . . . 6  |-  ( B  e.  NN  ->  0  <  B )
134, 12jca 518 . . . . 5  |-  ( B  e.  NN  ->  ( B  e.  RR  /\  0  <  B ) )
14 divge0 9641 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
1511, 13, 14syl2an 463 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  0  <_  ( A  /  B ) )
16 flge0nn0 10964 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  -> 
( |_ `  ( A  /  B ) )  e.  NN0 )
179, 15, 16syl2anc 642 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  e.  NN0 )
181, 17syl5eqel 2380 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  NN0 )
19 quorem.2 . . 3  |-  R  =  ( A  -  ( B  x.  Q )
)
20 nnnn0 9988 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  NN0 )
2120adantl 452 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  NN0 )
22 nn0mulcl 10016 . . . . 5  |-  ( ( B  e.  NN0  /\  Q  e.  NN0 )  -> 
( B  x.  Q
)  e.  NN0 )
2321, 18, 22syl2anc 642 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  NN0 )
24 simpl 443 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  NN0 )
25 nn0cn 9991 . . . . . . . 8  |-  ( Q  e.  NN0  ->  Q  e.  CC )
2618, 25syl 15 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  e.  CC )
27 nncn 9770 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
2827adantl 452 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  B  e.  CC )
29 divcan3 9464 . . . . . . 7  |-  ( ( Q  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( B  x.  Q
)  /  B )  =  Q )
3026, 28, 7, 29syl3anc 1182 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  =  Q )
31 flle 10947 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  <_ 
( A  /  B
) )
329, 31syl 15 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( |_ `  ( A  /  B ) )  <_  ( A  /  B ) )
331, 32syl5eqbr 4072 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  Q  <_  ( A  /  B ) )
3430, 33eqbrtrd 4059 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  /  B
)  <_  ( A  /  B ) )
35 nn0re 9990 . . . . . . 7  |-  ( ( B  x.  Q )  e.  NN0  ->  ( B  x.  Q )  e.  RR )
3623, 35syl 15 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  RR )
3712adantl 452 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  0  <  B )
38 lediv1 9637 . . . . . 6  |-  ( ( ( B  x.  Q
)  e.  RR  /\  A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
3936, 3, 5, 37, 38syl112anc 1186 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  <_  A  <->  ( ( B  x.  Q
)  /  B )  <_  ( A  /  B ) ) )
4034, 39mpbird 223 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  <_  A )
41 nn0sub2 10093 . . . 4  |-  ( ( ( B  x.  Q
)  e.  NN0  /\  A  e.  NN0  /\  ( B  x.  Q )  <_  A )  ->  ( A  -  ( B  x.  Q ) )  e. 
NN0 )
4223, 24, 40, 41syl3anc 1182 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  -  ( B  x.  Q )
)  e.  NN0 )
4319, 42syl5eqel 2380 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  NN0 )
441oveq2i 5885 . . . . . 6  |-  ( ( A  /  B )  -  Q )  =  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )
45 fraclt1 10950 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
469, 45syl 15 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
4744, 46syl5eqbr 4072 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  Q
)  <  1 )
4819oveq1i 5884 . . . . . 6  |-  ( R  /  B )  =  ( ( A  -  ( B  x.  Q
) )  /  B
)
49 nn0cn 9991 . . . . . . . . 9  |-  ( A  e.  NN0  ->  A  e.  CC )
5049adantr 451 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  e.  CC )
51 nn0cn 9991 . . . . . . . . 9  |-  ( ( B  x.  Q )  e.  NN0  ->  ( B  x.  Q )  e.  CC )
5223, 51syl 15 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  x.  Q
)  e.  CC )
5327, 6jca 518 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
5453adantl 452 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
55 divsubdir 9472 . . . . . . . 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 ) ) )
5650, 52, 54, 55syl3anc 1182 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  ( ( B  x.  Q )  /  B ) ) )
5730oveq2d 5890 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  /  B )  -  (
( B  x.  Q
)  /  B ) )  =  ( ( A  /  B )  -  Q ) )
5856, 57eqtrd 2328 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( A  -  ( B  x.  Q
) )  /  B
)  =  ( ( A  /  B )  -  Q ) )
5948, 58syl5eq 2340 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  =  ( ( A  /  B )  -  Q ) )
60 divid 9467 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
6127, 6, 60syl2anc 642 . . . . . 6  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
6261adantl 452 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  /  B
)  =  1 )
6347, 59, 623brtr4d 4069 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  /  B
)  <  ( B  /  B ) )
64 nn0re 9990 . . . . . 6  |-  ( R  e.  NN0  ->  R  e.  RR )
6543, 64syl 15 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  e.  RR )
66 ltdiv1 9636 . . . . 5  |-  ( ( R  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
6765, 5, 5, 37, 66syl112anc 1186 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  <  B  <->  ( R  /  B )  <  ( B  /  B ) ) )
6863, 67mpbird 223 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  R  <  B )
6919oveq2i 5885 . . . 4  |-  ( ( B  x.  Q )  +  R )  =  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )
70 pncan3 9075 . . . . 5  |-  ( ( ( B  x.  Q
)  e.  CC  /\  A  e.  CC )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
7152, 50, 70syl2anc 642 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( B  x.  Q )  +  ( A  -  ( B  x.  Q ) ) )  =  A )
7269, 71syl5req 2341 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  A  =  ( ( B  x.  Q )  +  R ) )
7368, 72jca 518 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( R  <  B  /\  A  =  (
( B  x.  Q
)  +  R ) ) )
7418, 43, 73jca31 520 1  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e. 
NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   |_cfl 10940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fl 10941
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