Theorem vdwlem5 15689

Description: Lemma for vdw 15698. (Contributed by Mario Carneiro, 12-Sep-2014.)

Hypotheses

Ref	Expression
vdwlem3.v	|- ( ph -> V e. NN )
vdwlem3.w	|- ( ph -> W e. NN )
vdwlem4.r	|- ( ph -> R e. Fin )
vdwlem4.h	|- ( ph -> H : ( 1 ... ( W x. ( 2 x. V ) ) ) --> R )
vdwlem4.f	|- F = ( x e. ( 1 ... V ) |-> ( y e. ( 1 ... W ) |-> ( H ` ( y + ( W x. ( ( x - 1 ) + V ) ) ) ) ) )
vdwlem7.m	|- ( ph -> M e. NN )
vdwlem7.g	|- ( ph -> G : ( 1 ... W ) --> R )
vdwlem7.k	|- ( ph -> K e. ( ZZ>= ` 2 ) )
vdwlem7.a	|- ( ph -> A e. NN )
vdwlem7.d	|- ( ph -> D e. NN )
vdwlem7.s	|- ( ph -> ( A (AP ` K ) D ) C_ ( `' F " { G } ) )
vdwlem6.b	|- ( ph -> B e. NN )
vdwlem6.e	|- ( ph -> E : ( 1 ... M ) --> NN )
vdwlem6.s	|- ( ph -> A. i e. ( 1 ... M ) ( ( B + ( E ` i ) ) (AP ` K ) ( E ` i ) ) C_ ( `' G " { ( G ` ( B + ( E ` i ) ) ) } ) )
vdwlem6.j	|- J = ( i e. ( 1 ... M ) |-> ( G ` ( B + ( E ` i ) ) ) )
vdwlem6.r	|- ( ph -> ( # ` ran J ) = M )
vdwlem6.t	|- T = ( B + ( W x. ( ( A + ( V - D ) ) - 1 ) ) )
vdwlem6.p	|- P = ( j e. ( 1 ... ( M + 1 ) ) |-> ( if ( j = ( M + 1 ) , 0 , ( E ` j ) ) + ( W x. D ) ) )

Assertion

Ref	Expression
vdwlem5	|- ( ph -> T e. NN )

Distinct variable groups:   x, y, A   i, j, x, y, G   i, K, j, x, y   i, J, j, x   P, i, x   ph, i, j, x, y   R, i, x, y   B, i, j, x, y   i, H, x, y   i, M, j, x, y   D, j, x, y   i, E, j, x, y   i, W, j, x, y   T, i, x   x, V, y

Allowed substitution hints:   A( i, j)   D( i)   P( y, j)   R( j)   T( y, j)   F( x, y, i, j)   H( j)   J( y)   V( i, j)

Proof of Theorem vdwlem5

Step	Hyp	Ref	Expression
1		vdwlem6.t	. 2 |- T = ( B + ( W x. ( ( A + ( V - D ) ) - 1 ) ) )
2		vdwlem6.b	. . 3 |- ( ph -> B e. NN )
3		vdwlem3.w	. . . . 5 |- ( ph -> W e. NN )
4	3	nnnn0d 11351	. . . 4 |- ( ph -> W e. NN0 )
5		vdwlem7.a	. . . . . 6 |- ( ph -> A e. NN )
6		vdwlem3.v	. . . . . . . . . 10 |- ( ph -> V e. NN )
7	6	nncnd 11036	. . . . . . . . 9 |- ( ph -> V e. CC )
8		vdwlem7.d	. . . . . . . . . 10 |- ( ph -> D e. NN )
9	8	nncnd 11036	. . . . . . . . 9 |- ( ph -> D e. CC )
10	7, 9	subcld 10392	. . . . . . . 8 |- ( ph -> ( V - D ) e. CC )
11	5	nncnd 11036	. . . . . . . 8 |- ( ph -> A e. CC )
12	10, 11	npcand 10396	. . . . . . 7 |- ( ph -> ( ( ( V - D ) - A ) + A ) = ( V - D ) )
13	7, 9, 11	subsub4d 10423	. . . . . . . . . 10 |- ( ph -> ( ( V - D ) - A ) = ( V - ( D + A ) ) )
14	9, 11	addcomd 10238	. . . . . . . . . . 11 |- ( ph -> ( D + A ) = ( A + D ) )
15	14	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( V - ( D + A ) ) = ( V - ( A + D ) ) )
16	13, 15	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( V - D ) - A ) = ( V - ( A + D ) ) )
17		cnvimass 5485	. . . . . . . . . . . . 13 |- ( `' F " { G } ) C_ dom F
18		vdwlem4.r	. . . . . . . . . . . . . . 15 |- ( ph -> R e. Fin )
19		vdwlem4.h	. . . . . . . . . . . . . . 15 |- ( ph -> H : ( 1 ... ( W x. ( 2 x. V ) ) ) --> R )
20		vdwlem4.f	. . . . . . . . . . . . . . 15 |- F = ( x e. ( 1 ... V ) |-> ( y e. ( 1 ... W ) |-> ( H ` ( y + ( W x. ( ( x - 1 ) + V ) ) ) ) ) )
21	6, 3, 18, 19, 20	vdwlem4 15688	. . . . . . . . . . . . . 14 |- ( ph -> F : ( 1 ... V ) --> ( R ^m ( 1 ... W ) ) )
22		fdm 6051	. . . . . . . . . . . . . 14 |- ( F : ( 1 ... V ) --> ( R ^m ( 1 ... W ) ) -> dom F = ( 1 ... V ) )
23	21, 22	syl 17	. . . . . . . . . . . . 13 |- ( ph -> dom F = ( 1 ... V ) )
24	17, 23	syl5sseq 3653	. . . . . . . . . . . 12 |- ( ph -> ( `' F " { G } ) C_ ( 1 ... V ) )
25		vdwlem7.s	. . . . . . . . . . . . 13 |- ( ph -> ( A (AP ` K ) D ) C_ ( `' F " { G } ) )
26		ssun2 3777	. . . . . . . . . . . . . . 15 |- ( ( A + D ) (AP ` ( K - 1 ) ) D ) C_ ( { A } u. ( ( A + D ) (AP ` ( K - 1 ) ) D ) )
27		vdwlem7.k	. . . . . . . . . . . . . . . . 17 |- ( ph -> K e. ( ZZ>= ` 2 ) )
28		uz2m1nn 11763	. . . . . . . . . . . . . . . . 17 |- ( K e. ( ZZ>= ` 2 ) -> ( K - 1 ) e. NN )
29	27, 28	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( K - 1 ) e. NN )
30	5, 8	nnaddcld 11067	. . . . . . . . . . . . . . . 16 |- ( ph -> ( A + D ) e. NN )
31		vdwapid1 15679	. . . . . . . . . . . . . . . 16 |- ( ( ( K - 1 ) e. NN /\ ( A + D ) e. NN /\ D e. NN ) -> ( A + D ) e. ( ( A + D ) (AP ` ( K - 1 ) ) D ) )
32	29, 30, 8, 31	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ph -> ( A + D ) e. ( ( A + D ) (AP ` ( K - 1 ) ) D ) )
33	26, 32	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ph -> ( A + D ) e. ( { A } u. ( ( A + D ) (AP ` ( K - 1 ) ) D ) ) )
34		eluz2nn 11726	. . . . . . . . . . . . . . . . . . . 20 |- ( K e. ( ZZ>= ` 2 ) -> K e. NN )
35	27, 34	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> K e. NN )
36	35	nncnd 11036	. . . . . . . . . . . . . . . . . 18 |- ( ph -> K e. CC )
37		ax-1cn 9994	. . . . . . . . . . . . . . . . . 18 |- 1 e. CC
38		npcan 10290	. . . . . . . . . . . . . . . . . 18 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K - 1 ) + 1 ) = K )
39	36, 37, 38	sylancl 694	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( K - 1 ) + 1 ) = K )
40	39	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ph -> (AP ` ( ( K - 1 ) + 1 ) ) = (AP ` K ) )
41	40	oveqd 6667	. . . . . . . . . . . . . . 15 |- ( ph -> ( A (AP ` ( ( K - 1 ) + 1 ) ) D ) = ( A (AP ` K ) D ) )
42		nnm1nn0 11334	. . . . . . . . . . . . . . . . 17 |- ( K e. NN -> ( K - 1 ) e. NN0 )
43	35, 42	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( K - 1 ) e. NN0 )
44		vdwapun 15678	. . . . . . . . . . . . . . . 16 |- ( ( ( K - 1 ) e. NN0 /\ A e. NN /\ D e. NN ) -> ( A (AP ` ( ( K - 1 ) + 1 ) ) D ) = ( { A } u. ( ( A + D ) (AP ` ( K - 1 ) ) D ) ) )
45	43, 5, 8, 44	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ph -> ( A (AP ` ( ( K - 1 ) + 1 ) ) D ) = ( { A } u. ( ( A + D ) (AP ` ( K - 1 ) ) D ) ) )
46	41, 45	eqtr3d 2658	. . . . . . . . . . . . . 14 |- ( ph -> ( A (AP ` K ) D ) = ( { A } u. ( ( A + D ) (AP ` ( K - 1 ) ) D ) ) )
47	33, 46	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ph -> ( A + D ) e. ( A (AP ` K ) D ) )
48	25, 47	sseldd 3604	. . . . . . . . . . . 12 |- ( ph -> ( A + D ) e. ( `' F " { G } ) )
49	24, 48	sseldd 3604	. . . . . . . . . . 11 |- ( ph -> ( A + D ) e. ( 1 ... V ) )
50		elfzuz3 12339	. . . . . . . . . . 11 |- ( ( A + D ) e. ( 1 ... V ) -> V e. ( ZZ>= ` ( A + D ) ) )
51	49, 50	syl 17	. . . . . . . . . 10 |- ( ph -> V e. ( ZZ>= ` ( A + D ) ) )
52		uznn0sub 11719	. . . . . . . . . 10 |- ( V e. ( ZZ>= ` ( A + D ) ) -> ( V - ( A + D ) ) e. NN0 )
53	51, 52	syl 17	. . . . . . . . 9 |- ( ph -> ( V - ( A + D ) ) e. NN0 )
54	16, 53	eqeltrd 2701	. . . . . . . 8 |- ( ph -> ( ( V - D ) - A ) e. NN0 )
55		nn0nnaddcl 11324	. . . . . . . 8 |- ( ( ( ( V - D ) - A ) e. NN0 /\ A e. NN ) -> ( ( ( V - D ) - A ) + A ) e. NN )
56	54, 5, 55	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( ( V - D ) - A ) + A ) e. NN )
57	12, 56	eqeltrrd 2702	. . . . . 6 |- ( ph -> ( V - D ) e. NN )
58	5, 57	nnaddcld 11067	. . . . 5 |- ( ph -> ( A + ( V - D ) ) e. NN )
59		nnm1nn0 11334	. . . . 5 |- ( ( A + ( V - D ) ) e. NN -> ( ( A + ( V - D ) ) - 1 ) e. NN0 )
60	58, 59	syl 17	. . . 4 |- ( ph -> ( ( A + ( V - D ) ) - 1 ) e. NN0 )
61	4, 60	nn0mulcld 11356	. . 3 |- ( ph -> ( W x. ( ( A + ( V - D ) ) - 1 ) ) e. NN0 )
62		nnnn0addcl 11323	. . 3 |- ( ( B e. NN /\ ( W x. ( ( A + ( V - D ) ) - 1 ) ) e. NN0 ) -> ( B + ( W x. ( ( A + ( V - D ) ) - 1 ) ) ) e. NN )
63	2, 61, 62	syl2anc 693	. 2 |- ( ph -> ( B + ( W x. ( ( A + ( V - D ) ) - 1 ) ) ) e. NN )
64	1, 63	syl5eqel 2705	1 |- ( ph -> T e. NN )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   #chash 13117  APcvdwa 15669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-vdwap 15672

This theorem is referenced by:  vdwlem6  15690