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Theorem vdwlem3 14155
Description: Lemma for vdw 14166. (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 11588 . . . . . 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 11588 . . . . . . . . 9  |-  ( A  e.  ( 1 ... V )  ->  A  e.  NN )
75, 6syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  NN )
8 nnm1nn0 10725 . . . . . . . 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 10715 . . . . . . 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 10473 . . . . 5  |-  ( ph  ->  ( W  x.  (
( A  -  1 )  +  V ) )  e.  NN )
143, 13nnaddcld 10472 . . . 4  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  NN )
1514nnred 10441 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  RR )
167, 10nnaddcld 10472 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  NN )
174, 16nnmulcld 10473 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  NN )
1817nnred 10441 . . 3  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  e.  RR )
19 2nn 10583 . . . . . 6  |-  2  e.  NN
20 nnmulcl 10449 . . . . . 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 10473 . . . 4  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  NN )
2322nnred 10441 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  RR )
24 elfzle2 11565 . . . . . 6  |-  ( B  e.  ( 1 ... W )  ->  B  <_  W )
251, 24syl 16 . . . . 5  |-  ( ph  ->  B  <_  W )
26 nnre 10433 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  RR )
27 nnre 10433 . . . . . . 7  |-  ( W  e.  NN  ->  W  e.  RR )
28 nnre 10433 . . . . . . 7  |-  ( ( W  x.  ( ( A  -  1 )  +  V ) )  e.  NN  ->  ( W  x.  ( ( A  -  1 )  +  V ) )  e.  RR )
29 leadd1 9911 . . . . . . 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 1261 . . . . . 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 1219 . . . . 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 10442 . . . . . 6  |-  ( ph  ->  W  e.  CC )
34 ax-1cn 9444 . . . . . . 7  |-  1  e.  CC
3534a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
3612nncnd 10442 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  +  V
)  e.  CC )
3733, 35, 36adddid 9514 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( ( W  x.  1 )  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
389nn0cnd 10742 . . . . . . . 8  |-  ( ph  ->  ( A  -  1 )  e.  CC )
3910nncnd 10442 . . . . . . . 8  |-  ( ph  ->  V  e.  CC )
4035, 38, 39addassd 9512 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( 1  +  ( ( A  -  1 )  +  V ) ) )
417nncnd 10442 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
42 pncan3 9722 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  ( A  -  1 ) )  =  A )
4334, 41, 42sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  ( A  -  1 ) )  =  A )
4443oveq1d 6208 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( A  -  1 ) )  +  V
)  =  ( A  +  V ) )
4540, 44eqtr3d 2494 . . . . . 6  |-  ( ph  ->  ( 1  +  ( ( A  -  1 )  +  V ) )  =  ( A  +  V ) )
4645oveq2d 6209 . . . . 5  |-  ( ph  ->  ( W  x.  (
1  +  ( ( A  -  1 )  +  V ) ) )  =  ( W  x.  ( A  +  V ) ) )
4733mulid1d 9507 . . . . . 6  |-  ( ph  ->  ( W  x.  1 )  =  W )
4847oveq1d 6208 . . . . 5  |-  ( ph  ->  ( ( W  x.  1 )  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
4937, 46, 483eqtr3d 2500 . . . 4  |-  ( ph  ->  ( W  x.  ( A  +  V )
)  =  ( W  +  ( W  x.  ( ( A  - 
1 )  +  V
) ) ) )
5032, 49breqtrrd 4419 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( A  +  V
) ) )
517nnred 10441 . . . . . 6  |-  ( ph  ->  A  e.  RR )
5210nnred 10441 . . . . . 6  |-  ( ph  ->  V  e.  RR )
53 elfzle2 11565 . . . . . . 7  |-  ( A  e.  ( 1 ... V )  ->  A  <_  V )
545, 53syl 16 . . . . . 6  |-  ( ph  ->  A  <_  V )
5551, 52, 52, 54leadd1dd 10057 . . . . 5  |-  ( ph  ->  ( A  +  V
)  <_  ( V  +  V ) )
56392timesd 10671 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  =  ( V  +  V ) )
5755, 56breqtrrd 4419 . . . 4  |-  ( ph  ->  ( A  +  V
)  <_  ( 2  x.  V ) )
5816nnred 10441 . . . . 5  |-  ( ph  ->  ( A  +  V
)  e.  RR )
5921nnred 10441 . . . . 5  |-  ( ph  ->  ( 2  x.  V
)  e.  RR )
604nnred 10441 . . . . 5  |-  ( ph  ->  W  e.  RR )
614nngt0d 10469 . . . . 5  |-  ( ph  ->  0  <  W )
62 lemul2 10286 . . . . 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 1223 . . . 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 9632 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  <_  ( W  x.  ( 2  x.  V
) ) )
66 nnuz 11000 . . . 4  |-  NN  =  ( ZZ>= `  1 )
6714, 66syl6eleq 2549 . . 3  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( ZZ>= ` 
1 ) )
6822nnzd 10850 . . 3  |-  ( ph  ->  ( W  x.  (
2  x.  V ) )  e.  ZZ )
69 elfz5 11555 . . 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 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391    < clt 9522    <_ cle 9523    - cmin 9699   NNcn 10426   2c2 10475   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548
This theorem is referenced by:  vdwlem4  14156  vdwlem6  14158
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