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Theorem vdwlem3 14036
Description: Lemma for vdw 14047. (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 11470 . . . . . 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 11470 . . . . . . . . 9  |-  ( A  e.  ( 1 ... V )  ->  A  e.  NN )
75, 6syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  NN )
8 nnm1nn0 10613 . . . . . . . 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 10603 . . . . . . 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 10361 . . . . 5  |-  ( ph  ->  ( W  x.  (
( A  -  1 )  +  V ) )  e.  NN )
143, 13nnaddcld 10360 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  NN )
1514nnred 10329 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  RR )
167, 10nnaddcld 10360 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  NN )
174, 16nnmulcld 10361 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  NN )
1817nnred 10329 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  RR )
19 2nn 10471 . . . . . 6  |-  2  e.  NN
20 nnmulcl 10337 . . . . . 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 10361 . . . 4  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  NN )
2322nnred 10329 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  RR )
24 elfzle2 11447 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  <_  W )
251, 24syl 16 . . . . 5  |-  ( ph  ->  B  <_  W )
26 nnre 10321 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
27 nnre 10321 . . . . . . 7  |-  ( W  e.  NN  ->  W  e.  RR )
28 nnre 10321 . . . . . . 7  |-  ( ( W  x.  ( ( A  -  1 )  +  V ) )  e.  NN  ->  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  RR )
29 leadd1 9799 . . . . . . 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 1260 . . . . . 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 1218 . . . . 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 10330 . . . . . 6  |-  ( ph  ->  W  e.  CC )
34 ax-1cn 9332 . . . . . . 7  |-  1  e.  CC
3534a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
3612nncnd 10330 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  +  V
)  e.  CC )
3733, 35, 36adddid 9402 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( ( W  x.  1 )  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
389nn0cnd 10630 . . . . . . . 8  |-  ( ph  ->  ( A  -  1 )  e.  CC )
3910nncnd 10330 . . . . . . . 8  |-  ( ph  ->  V  e.  CC )
4035, 38, 39addassd 9400 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( 1  +  ( ( A  -  1 )  +  V ) ) )
417nncnd 10330 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
42 pncan3 9610 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  ( A  -  1 ) )  =  A )
4334, 41, 42sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  ( A  -  1 ) )  =  A )
4443oveq1d 6101 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( A  +  V ) )
4540, 44eqtr3d 2472 . . . . . 6  |-  ( ph  ->  ( 1  +  ( ( A  -  1 )  +  V ) )  =  ( A  +  V ) )
4645oveq2d 6102 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( W  x.  ( A  +  V ) ) )
4733mulid1d 9395 . . . . . 6  |-  ( ph  ->  ( W  x.  1 )  =  W )
4847oveq1d 6101 . . . . 5  |-  ( ph  ->  ( ( W  x.  1 )  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
4937, 46, 483eqtr3d 2478 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
5032, 49breqtrrd 4313 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( A  +  V
) ) )
517nnred 10329 . . . . . 6  |-  ( ph  ->  A  e.  RR )
5210nnred 10329 . . . . . 6  |-  ( ph  ->  V  e.  RR )
53 elfzle2 11447 . . . . . . 7  |-  ( A  e.  ( 1 ... V )  ->  A  <_  V )
545, 53syl 16 . . . . . 6  |-  ( ph  ->  A  <_  V )
5551, 52, 52, 54leadd1dd 9945 . . . . 5  |-  ( ph  ->  ( A  +  V
)  <_  ( V  +  V ) )
56392timesd 10559 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  =  ( V  +  V ) )
5755, 56breqtrrd 4313 . . . 4  |-  ( ph  ->  ( A  +  V
)  <_  ( 2  x.  V ) )
5816nnred 10329 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  RR )
5921nnred 10329 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  e.  RR )
604nnred 10329 . . . . 5  |-  ( ph  ->  W  e.  RR )
614nngt0d 10357 . . . . 5  |-  ( ph  ->  0  <  W )
62 lemul2 10174 . . . . 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 1222 . . . 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 9520 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) )
66 nnuz 10888 . . . 4  |-  NN  =  ( ZZ>= `  1 )
6714, 66syl6eleq 2528 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( ZZ>= ` 
1 ) )
6822nnzd 10738 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  ZZ )
69 elfz5 11437 . . 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 1369    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430
This theorem is referenced by:  vdwlem4  14037  vdwlem6  14039
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