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Theorem vdwlem3 14712
Description: Lemma for vdw 14723. (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 11770 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  e.  NN )
31, 2syl 17 . . . . 5  |-  ( ph  ->  B  e.  NN )
4 vdwlem3.w . . . . . 6  |-  ( ph  ->  W  e.  NN )
5 vdwlem3.a . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 1 ... V ) )
6 elfznn 11770 . . . . . . . . 9  |-  ( A  e.  ( 1 ... V )  ->  A  e.  NN )
75, 6syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  NN )
8 nnm1nn0 10880 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  -  1 )  e.  NN0 )
97, 8syl 17 . . . . . . 7  |-  ( ph  ->  ( A  -  1 )  e.  NN0 )
10 vdwlem3.v . . . . . . 7  |-  ( ph  ->  V  e.  NN )
11 nn0nnaddcl 10870 . . . . . . 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 10626 . . . . 5  |-  ( ph  ->  ( W  x.  (
( A  -  1 )  +  V ) )  e.  NN )
143, 13nnaddcld 10625 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  NN )
1514nnred 10593 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  RR )
167, 10nnaddcld 10625 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  NN )
174, 16nnmulcld 10626 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  NN )
1817nnred 10593 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  RR )
19 2nn 10736 . . . . . 6  |-  2  e.  NN
20 nnmulcl 10601 . . . . . 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 10626 . . . 4  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  NN )
2322nnred 10593 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  RR )
24 elfzle2 11746 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  <_  W )
251, 24syl 17 . . . . 5  |-  ( ph  ->  B  <_  W )
26 nnre 10585 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
27 nnre 10585 . . . . . . 7  |-  ( W  e.  NN  ->  W  e.  RR )
28 nnre 10585 . . . . . . 7  |-  ( ( W  x.  ( ( A  -  1 )  +  V ) )  e.  NN  ->  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  RR )
29 leadd1 10063 . . . . . . 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 1274 . . . . . 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 1232 . . . . 5  |-  ( ph  ->  ( B  <_  W  <->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  (
( A  -  1 )  +  V ) ) ) ) )
3225, 31mpbid 212 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
334nncnd 10594 . . . . . 6  |-  ( ph  ->  W  e.  CC )
34 1cnd 9644 . . . . . 6  |-  ( ph  ->  1  e.  CC )
3512nncnd 10594 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  +  V
)  e.  CC )
3633, 34, 35adddid 9652 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( ( W  x.  1 )  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
379nn0cnd 10897 . . . . . . . 8  |-  ( ph  ->  ( A  -  1 )  e.  CC )
3810nncnd 10594 . . . . . . . 8  |-  ( ph  ->  V  e.  CC )
3934, 37, 38addassd 9650 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( 1  +  ( ( A  -  1 )  +  V ) ) )
40 ax-1cn 9582 . . . . . . . . 9  |-  1  e.  CC
417nncnd 10594 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
42 pncan3 9866 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  ( A  -  1 ) )  =  A )
4340, 41, 42sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  ( A  -  1 ) )  =  A )
4443oveq1d 6295 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( A  +  V ) )
4539, 44eqtr3d 2447 . . . . . 6  |-  ( ph  ->  ( 1  +  ( ( A  -  1 )  +  V ) )  =  ( A  +  V ) )
4645oveq2d 6296 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( W  x.  ( A  +  V ) ) )
4733mulid1d 9645 . . . . . 6  |-  ( ph  ->  ( W  x.  1 )  =  W )
4847oveq1d 6295 . . . . 5  |-  ( ph  ->  ( ( W  x.  1 )  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
4936, 46, 483eqtr3d 2453 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
5032, 49breqtrrd 4423 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( A  +  V
) ) )
517nnred 10593 . . . . . 6  |-  ( ph  ->  A  e.  RR )
5210nnred 10593 . . . . . 6  |-  ( ph  ->  V  e.  RR )
53 elfzle2 11746 . . . . . . 7  |-  ( A  e.  ( 1 ... V )  ->  A  <_  V )
545, 53syl 17 . . . . . 6  |-  ( ph  ->  A  <_  V )
5551, 52, 52, 54leadd1dd 10208 . . . . 5  |-  ( ph  ->  ( A  +  V
)  <_  ( V  +  V ) )
56382timesd 10824 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  =  ( V  +  V ) )
5755, 56breqtrrd 4423 . . . 4  |-  ( ph  ->  ( A  +  V
)  <_  ( 2  x.  V ) )
5816nnred 10593 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  RR )
5921nnred 10593 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  e.  RR )
604nnred 10593 . . . . 5  |-  ( ph  ->  W  e.  RR )
614nngt0d 10622 . . . . 5  |-  ( ph  ->  0  <  W )
62 lemul2 10438 . . . . 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 1236 . . . 4  |-  ( ph  ->  ( ( A  +  V )  <_  (
2  x.  V )  <-> 
( W  x.  ( A  +  V )
)  <_  ( W  x.  ( 2  x.  V
) ) ) )
6457, 63mpbid 212 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  <_  ( W  x.  ( 2  x.  V
) ) )
6515, 18, 23, 50, 64letrd 9775 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) )
66 nnuz 11164 . . . 4  |-  NN  =  ( ZZ>= `  1 )
6714, 66syl6eleq 2502 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( ZZ>= ` 
1 ) )
6822nnzd 11009 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  ZZ )
69 elfz5 11736 . . 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 234 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 186    = wceq 1407    e. wcel 1844   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   CCcc 9522   RRcr 9523   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    < clt 9660    <_ cle 9661    - cmin 9843   NNcn 10578   2c2 10628   NN0cn0 10838   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729
This theorem is referenced by:  vdwlem4  14713  vdwlem6  14715
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