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

Step	Hyp	Ref	Expression
1		vdwlem3.b	. . . . . 6 |- ( ph -> B e. ( 1 ... W ) )
2		elfznn 11588	. . . . . 6 |- ( B e. ( 1 ... W ) -> B e. NN )
3	1, 2	syl 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 )
7	5, 6	syl 16	. . . . . . . 8 |- ( ph -> A e. NN )
8		nnm1nn0 10725	. . . . . . . 8 |- ( A e. NN -> ( A - 1 ) e. NN0 )
9	7, 8	syl 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 )
12	9, 10, 11	syl2anc 661	. . . . . 6 |- ( ph -> ( ( A - 1 ) + V ) e. NN )
13	4, 12	nnmulcld 10473	. . . . 5 |- ( ph -> ( W x. ( ( A - 1 ) + V ) ) e. NN )
14	3, 13	nnaddcld 10472	. . . 4 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) e. NN )
15	14	nnred 10441	. . 3 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) e. RR )
16	7, 10	nnaddcld 10472	. . . . 5 |- ( ph -> ( A + V ) e. NN )
17	4, 16	nnmulcld 10473	. . . 4 |- ( ph -> ( W x. ( A + V ) ) e. NN )
18	17	nnred 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 )
21	19, 10, 20	sylancr 663	. . . . 5 |- ( ph -> ( 2 x. V ) e. NN )
22	4, 21	nnmulcld 10473	. . . 4 |- ( ph -> ( W x. ( 2 x. V ) ) e. NN )
23	22	nnred 10441	. . 3 |- ( ph -> ( W x. ( 2 x. V ) ) e. RR )
24		elfzle2 11565	. . . . . 6 |- ( B e. ( 1 ... W ) -> B <_ W )
25	1, 24	syl 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 ) ) ) ) )
30	26, 27, 28, 29	syl3an 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 ) ) ) ) )
31	3, 4, 13, 30	syl3anc 1219	. . . . 5 |- ( ph -> ( B <_ W <-> ( B + ( W x. ( ( A - 1 ) + V ) ) ) <_ ( W + ( W x. ( ( A - 1 ) + V ) ) ) ) )
32	25, 31	mpbid 210	. . . 4 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) <_ ( W + ( W x. ( ( A - 1 ) + V ) ) ) )
33	4	nncnd 10442	. . . . . 6 |- ( ph -> W e. CC )
34		ax-1cn 9444	. . . . . . 7 |- 1 e. CC
35	34	a1i 11	. . . . . 6 |- ( ph -> 1 e. CC )
36	12	nncnd 10442	. . . . . 6 |- ( ph -> ( ( A - 1 ) + V ) e. CC )
37	33, 35, 36	adddid 9514	. . . . 5 |- ( ph -> ( W x. ( 1 + ( ( A - 1 ) + V ) ) ) = ( ( W x. 1 ) + ( W x. ( ( A - 1 ) + V ) ) ) )
38	9	nn0cnd 10742	. . . . . . . 8 |- ( ph -> ( A - 1 ) e. CC )
39	10	nncnd 10442	. . . . . . . 8 |- ( ph -> V e. CC )
40	35, 38, 39	addassd 9512	. . . . . . 7 |- ( ph -> ( ( 1 + ( A - 1 ) ) + V ) = ( 1 + ( ( A - 1 ) + V ) ) )
41	7	nncnd 10442	. . . . . . . . 9 |- ( ph -> A e. CC )
42		pncan3 9722	. . . . . . . . 9 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + ( A - 1 ) ) = A )
43	34, 41, 42	sylancr 663	. . . . . . . 8 |- ( ph -> ( 1 + ( A - 1 ) ) = A )
44	43	oveq1d 6208	. . . . . . 7 |- ( ph -> ( ( 1 + ( A - 1 ) ) + V ) = ( A + V ) )
45	40, 44	eqtr3d 2494	. . . . . 6 |- ( ph -> ( 1 + ( ( A - 1 ) + V ) ) = ( A + V ) )
46	45	oveq2d 6209	. . . . 5 |- ( ph -> ( W x. ( 1 + ( ( A - 1 ) + V ) ) ) = ( W x. ( A + V ) ) )
47	33	mulid1d 9507	. . . . . 6 |- ( ph -> ( W x. 1 ) = W )
48	47	oveq1d 6208	. . . . 5 |- ( ph -> ( ( W x. 1 ) + ( W x. ( ( A - 1 ) + V ) ) ) = ( W + ( W x. ( ( A - 1 ) + V ) ) ) )
49	37, 46, 48	3eqtr3d 2500	. . . 4 |- ( ph -> ( W x. ( A + V ) ) = ( W + ( W x. ( ( A - 1 ) + V ) ) ) )
50	32, 49	breqtrrd 4419	. . 3 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) <_ ( W x. ( A + V ) ) )
51	7	nnred 10441	. . . . . 6 |- ( ph -> A e. RR )
52	10	nnred 10441	. . . . . 6 |- ( ph -> V e. RR )
53		elfzle2 11565	. . . . . . 7 |- ( A e. ( 1 ... V ) -> A <_ V )
54	5, 53	syl 16	. . . . . 6 |- ( ph -> A <_ V )
55	51, 52, 52, 54	leadd1dd 10057	. . . . 5 |- ( ph -> ( A + V ) <_ ( V + V ) )
56	39	2timesd 10671	. . . . 5 |- ( ph -> ( 2 x. V ) = ( V + V ) )
57	55, 56	breqtrrd 4419	. . . 4 |- ( ph -> ( A + V ) <_ ( 2 x. V ) )
58	16	nnred 10441	. . . . 5 |- ( ph -> ( A + V ) e. RR )
59	21	nnred 10441	. . . . 5 |- ( ph -> ( 2 x. V ) e. RR )
60	4	nnred 10441	. . . . 5 |- ( ph -> W e. RR )
61	4	nngt0d 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 ) ) ) )
63	58, 59, 60, 61, 62	syl112anc 1223	. . . 4 |- ( ph -> ( ( A + V ) <_ ( 2 x. V ) <-> ( W x. ( A + V ) ) <_ ( W x. ( 2 x. V ) ) ) )
64	57, 63	mpbid 210	. . 3 |- ( ph -> ( W x. ( A + V ) ) <_ ( W x. ( 2 x. V ) ) )
65	15, 18, 23, 50, 64	letrd 9632	. 2 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) <_ ( W x. ( 2 x. V ) ) )
66		nnuz 11000	. . . 4 |- NN = ( ZZ>= ` 1 )
67	14, 66	syl6eleq 2549	. . 3 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) e. ( ZZ>= ` 1 ) )
68	22	nnzd 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 ) ) ) )
70	67, 68, 69	syl2anc 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 ) ) ) )
71	65, 70	mpbird 232	1 |- ( ph -> ( B + ( W x. ( ( A - 1 ) + V ) ) ) e. ( 1 ... ( W x. ( 2 x. V ) ) ) )

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