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Theorem vdwlem3 14356
Description: Lemma for vdw 14367. (Contributed by Mario Carneiro, 13-Sep-2014.)
Hypotheses
Ref Expression
vdwlem3.v  |-  ( ph  ->  V  e.  NN )
vdwlem3.w  |-  ( ph  ->  W  e.  NN )
vdwlem3.a  |-  ( ph  ->  A  e.  ( 1 ... V ) )
vdwlem3.b  |-  ( ph  ->  B  e.  ( 1 ... W ) )
Assertion
Ref Expression
vdwlem3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( 1 ... ( W  x.  ( 2  x.  V
) ) ) )

Proof of Theorem vdwlem3
StepHypRef Expression
1 vdwlem3.b . . . . . 6  |-  ( ph  ->  B  e.  ( 1 ... W ) )
2 elfznn 11710 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  e.  NN )
31, 2syl 16 . . . . 5  |-  ( ph  ->  B  e.  NN )
4 vdwlem3.w . . . . . 6  |-  ( ph  ->  W  e.  NN )
5 vdwlem3.a . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 1 ... V ) )
6 elfznn 11710 . . . . . . . . 9  |-  ( A  e.  ( 1 ... V )  ->  A  e.  NN )
75, 6syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  NN )
8 nnm1nn0 10833 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  ( A  -  1 )  e.  NN0 )
10 vdwlem3.v . . . . . . 7  |-  ( ph  ->  V  e.  NN )
11 nn0nnaddcl 10823 . . . . . . 7  |-  ( ( ( A  -  1 )  e.  NN0  /\  V  e.  NN )  ->  ( ( A  - 
1 )  +  V
)  e.  NN )
129, 10, 11syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  +  V
)  e.  NN )
134, 12nnmulcld 10579 . . . . 5  |-  ( ph  ->  ( W  x.  (
( A  -  1 )  +  V ) )  e.  NN )
143, 13nnaddcld 10578 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  NN )
1514nnred 10547 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  RR )
167, 10nnaddcld 10578 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  NN )
174, 16nnmulcld 10579 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  NN )
1817nnred 10547 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  RR )
19 2nn 10689 . . . . . 6  |-  2  e.  NN
20 nnmulcl 10555 . . . . . 6  |-  ( ( 2  e.  NN  /\  V  e.  NN )  ->  ( 2  x.  V
)  e.  NN )
2119, 10, 20sylancr 663 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  e.  NN )
224, 21nnmulcld 10579 . . . 4  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  NN )
2322nnred 10547 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  RR )
24 elfzle2 11686 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  <_  W )
251, 24syl 16 . . . . 5  |-  ( ph  ->  B  <_  W )
26 nnre 10539 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
27 nnre 10539 . . . . . . 7  |-  ( W  e.  NN  ->  W  e.  RR )
28 nnre 10539 . . . . . . 7  |-  ( ( W  x.  ( ( A  -  1 )  +  V ) )  e.  NN  ->  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  RR )
29 leadd1 10016 . . . . . . 7  |-  ( ( B  e.  RR  /\  W  e.  RR  /\  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  RR )  -> 
( B  <_  W  <->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  (
( A  -  1 )  +  V ) ) ) ) )
3026, 27, 28, 29syl3an 1270 . . . . . 6  |-  ( ( B  e.  NN  /\  W  e.  NN  /\  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  NN )  -> 
( B  <_  W  <->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  (
( A  -  1 )  +  V ) ) ) ) )
313, 4, 13, 30syl3anc 1228 . . . . 5  |-  ( ph  ->  ( B  <_  W  <->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  (
( A  -  1 )  +  V ) ) ) ) )
3225, 31mpbid 210 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
334nncnd 10548 . . . . . 6  |-  ( ph  ->  W  e.  CC )
34 ax-1cn 9546 . . . . . . 7  |-  1  e.  CC
3534a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
3612nncnd 10548 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  +  V
)  e.  CC )
3733, 35, 36adddid 9616 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( ( W  x.  1 )  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
389nn0cnd 10850 . . . . . . . 8  |-  ( ph  ->  ( A  -  1 )  e.  CC )
3910nncnd 10548 . . . . . . . 8  |-  ( ph  ->  V  e.  CC )
4035, 38, 39addassd 9614 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( 1  +  ( ( A  -  1 )  +  V ) ) )
417nncnd 10548 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
42 pncan3 9824 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  ( A  -  1 ) )  =  A )
4334, 41, 42sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  ( A  -  1 ) )  =  A )
4443oveq1d 6297 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( A  +  V ) )
4540, 44eqtr3d 2510 . . . . . 6  |-  ( ph  ->  ( 1  +  ( ( A  -  1 )  +  V ) )  =  ( A  +  V ) )
4645oveq2d 6298 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( W  x.  ( A  +  V ) ) )
4733mulid1d 9609 . . . . . 6  |-  ( ph  ->  ( W  x.  1 )  =  W )
4847oveq1d 6297 . . . . 5  |-  ( ph  ->  ( ( W  x.  1 )  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
4937, 46, 483eqtr3d 2516 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
5032, 49breqtrrd 4473 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( A  +  V
) ) )
517nnred 10547 . . . . . 6  |-  ( ph  ->  A  e.  RR )
5210nnred 10547 . . . . . 6  |-  ( ph  ->  V  e.  RR )
53 elfzle2 11686 . . . . . . 7  |-  ( A  e.  ( 1 ... V )  ->  A  <_  V )
545, 53syl 16 . . . . . 6  |-  ( ph  ->  A  <_  V )
5551, 52, 52, 54leadd1dd 10162 . . . . 5  |-  ( ph  ->  ( A  +  V
)  <_  ( V  +  V ) )
56392timesd 10777 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  =  ( V  +  V ) )
5755, 56breqtrrd 4473 . . . 4  |-  ( ph  ->  ( A  +  V
)  <_  ( 2  x.  V ) )
5816nnred 10547 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  RR )
5921nnred 10547 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  e.  RR )
604nnred 10547 . . . . 5  |-  ( ph  ->  W  e.  RR )
614nngt0d 10575 . . . . 5  |-  ( ph  ->  0  <  W )
62 lemul2 10391 . . . . 5  |-  ( ( ( A  +  V
)  e.  RR  /\  ( 2  x.  V
)  e.  RR  /\  ( W  e.  RR  /\  0  <  W ) )  ->  ( ( A  +  V )  <_  ( 2  x.  V
)  <->  ( W  x.  ( A  +  V
) )  <_  ( W  x.  ( 2  x.  V ) ) ) )
6358, 59, 60, 61, 62syl112anc 1232 . . . 4  |-  ( ph  ->  ( ( A  +  V )  <_  (
2  x.  V )  <-> 
( W  x.  ( A  +  V )
)  <_  ( W  x.  ( 2  x.  V
) ) ) )
6457, 63mpbid 210 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  <_  ( W  x.  ( 2  x.  V
) ) )
6515, 18, 23, 50, 64letrd 9734 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) )
66 nnuz 11113 . . . 4  |-  NN  =  ( ZZ>= `  1 )
6714, 66syl6eleq 2565 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( ZZ>= ` 
1 ) )
6822nnzd 10961 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  ZZ )
69 elfz5 11676 . . 3  |-  ( ( ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( ZZ>= ` 
1 )  /\  ( W  x.  ( 2  x.  V ) )  e.  ZZ )  -> 
( ( B  +  ( W  x.  (
( A  -  1 )  +  V ) ) )  e.  ( 1 ... ( W  x.  ( 2  x.  V ) ) )  <-> 
( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) ) )
7067, 68, 69syl2anc 661 . 2  |-  ( ph  ->  ( ( B  +  ( W  x.  (
( A  -  1 )  +  V ) ) )  e.  ( 1 ... ( W  x.  ( 2  x.  V ) ) )  <-> 
( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) ) )
7165, 70mpbird 232 1  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( 1 ... ( W  x.  ( 2  x.  V
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625    - cmin 9801   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669
This theorem is referenced by:  vdwlem4  14357  vdwlem6  14359
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