Theorem vdwlem7 15691

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 } ) )

Assertion

Ref	Expression
vdwlem7	|- ( ph -> ( <. M , K >. PolyAP G -> ( <. ( M + 1 ) , K >. PolyAP H \/ ( K + 1 ) MonoAP G ) ) )

Distinct variable groups:   x, y, A   x, G, y   x, K, y   ph, x, y   x, R, y   x, H, y   x, M, y   x, D, y   x, W, y   x, V, y

Allowed substitution hints:   F( x, y)

Proof of Theorem vdwlem7

Dummy variables k a d i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( 1 ... W ) e. _V
2		2nn0 11309	. . . 4 |- 2 e. NN0
3		vdwlem7.k	. . . 4 |- ( ph -> K e. ( ZZ>= ` 2 ) )
4		eluznn0 11757	. . . 4 |- ( ( 2 e. NN0 /\ K e. ( ZZ>= ` 2 ) ) -> K e. NN0 )
5	2, 3, 4	sylancr 695	. . 3 |- ( ph -> K e. NN0 )
6		vdwlem7.g	. . 3 |- ( ph -> G : ( 1 ... W ) --> R )
7		vdwlem7.m	. . 3 |- ( ph -> M e. NN )
8		eqid 2622	. . 3 |- ( 1 ... M ) = ( 1 ... M )
9	1, 5, 6, 7, 8	vdwpc 15684	. 2 |- ( ph -> ( <. M , K >. PolyAP G <-> E. a e. NN E. d e. ( NN ^m ( 1 ... M ) ) ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
10		vdwlem3.v	. . . . . 6 |- ( ph -> V e. NN )
11	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> V e. NN )
12		vdwlem3.w	. . . . . 6 |- ( ph -> W e. NN )
13	12	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> W e. NN )
14		vdwlem4.r	. . . . . 6 |- ( ph -> R e. Fin )
15	14	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> R e. Fin )
16		vdwlem4.h	. . . . . 6 |- ( ph -> H : ( 1 ... ( W x. ( 2 x. V ) ) ) --> R )
17	16	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> H : ( 1 ... ( W x. ( 2 x. V ) ) ) --> R )
18		vdwlem4.f	. . . . 5 |- F = ( x e. ( 1 ... V ) |-> ( y e. ( 1 ... W ) |-> ( H ` ( y + ( W x. ( ( x - 1 ) + V ) ) ) ) ) )
19	7	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> M e. NN )
20	6	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> G : ( 1 ... W ) --> R )
21	3	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> K e. ( ZZ>= ` 2 ) )
22		vdwlem7.a	. . . . . 6 |- ( ph -> A e. NN )
23	22	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> A e. NN )
24		vdwlem7.d	. . . . . 6 |- ( ph -> D e. NN )
25	24	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> D e. NN )
26		vdwlem7.s	. . . . . 6 |- ( ph -> ( A (AP ` K ) D ) C_ ( `' F " { G } ) )
27	26	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> ( A (AP ` K ) D ) C_ ( `' F " { G } ) )
28		simplrl 800	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> a e. NN )
29		simplrr 801	. . . . . 6 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> d e. ( NN ^m ( 1 ... M ) ) )
30		nnex 11026	. . . . . . 7 |- NN e. _V
31		ovex 6678	. . . . . . 7 |- ( 1 ... M ) e. _V
32	30, 31	elmap 7886	. . . . . 6 |- ( d e. ( NN ^m ( 1 ... M ) ) <-> d : ( 1 ... M ) --> NN )
33	29, 32	sylib 208	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> d : ( 1 ... M ) --> NN )
34		simprl 794	. . . . . 6 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) )
35		fveq2 6191	. . . . . . . . . 10 |- ( i = k -> ( d ` i ) = ( d ` k ) )
36	35	oveq2d 6666	. . . . . . . . 9 |- ( i = k -> ( a + ( d ` i ) ) = ( a + ( d ` k ) ) )
37	36, 35	oveq12d 6668	. . . . . . . 8 |- ( i = k -> ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) = ( ( a + ( d ` k ) ) (AP ` K ) ( d ` k ) ) )
38	36	fveq2d 6195	. . . . . . . . . 10 |- ( i = k -> ( G ` ( a + ( d ` i ) ) ) = ( G ` ( a + ( d ` k ) ) ) )
39	38	sneqd 4189	. . . . . . . . 9 |- ( i = k -> { ( G ` ( a + ( d ` i ) ) ) } = { ( G ` ( a + ( d ` k ) ) ) } )
40	39	imaeq2d 5466	. . . . . . . 8 |- ( i = k -> ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) = ( `' G " { ( G ` ( a + ( d ` k ) ) ) } ) )
41	37, 40	sseq12d 3634	. . . . . . 7 |- ( i = k -> ( ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) <-> ( ( a + ( d ` k ) ) (AP ` K ) ( d ` k ) ) C_ ( `' G " { ( G ` ( a + ( d ` k ) ) ) } ) ) )
42	41	cbvralv 3171	. . . . . 6 |- ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) <-> A. k e. ( 1 ... M ) ( ( a + ( d ` k ) ) (AP ` K ) ( d ` k ) ) C_ ( `' G " { ( G ` ( a + ( d ` k ) ) ) } ) )
43	34, 42	sylib 208	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> A. k e. ( 1 ... M ) ( ( a + ( d ` k ) ) (AP ` K ) ( d ` k ) ) C_ ( `' G " { ( G ` ( a + ( d ` k ) ) ) } ) )
44	38	cbvmptv 4750	. . . . 5 |- ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) = ( k e. ( 1 ... M ) |-> ( G ` ( a + ( d ` k ) ) ) )
45		simprr 796	. . . . 5 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M )
46		eqid 2622	. . . . 5 |- ( a + ( W x. ( ( A + ( V - D ) ) - 1 ) ) ) = ( a + ( W x. ( ( A + ( V - D ) ) - 1 ) ) )
47		eqid 2622	. . . . 5 |- ( j e. ( 1 ... ( M + 1 ) ) |-> ( if ( j = ( M + 1 ) , 0 , ( d ` j ) ) + ( W x. D ) ) ) = ( j e. ( 1 ... ( M + 1 ) ) |-> ( if ( j = ( M + 1 ) , 0 , ( d ` j ) ) + ( W x. D ) ) )
48	11, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47	vdwlem6 15690	. . . 4 |- ( ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) /\ ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) ) -> ( <. ( M + 1 ) , K >. PolyAP H \/ ( K + 1 ) MonoAP G ) )
49	48	ex 450	. . 3 |- ( ( ph /\ ( a e. NN /\ d e. ( NN ^m ( 1 ... M ) ) ) ) -> ( ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) -> ( <. ( M + 1 ) , K >. PolyAP H \/ ( K + 1 ) MonoAP G ) ) )
50	49	rexlimdvva 3038	. 2 |- ( ph -> ( E. a e. NN E. d e. ( NN ^m ( 1 ... M ) ) ( A. i e. ( 1 ... M ) ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' G " { ( G ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... M ) |-> ( G ` ( a + ( d ` i ) ) ) ) ) = M ) -> ( <. ( M + 1 ) , K >. PolyAP H \/ ( K + 1 ) MonoAP G ) ) )
51	9, 50	sylbid 230	1 |- ( ph -> ( <. M , K >. PolyAP G -> ( <. ( M + 1 ) , K >. PolyAP H \/ ( K + 1 ) MonoAP G ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ifcif 4086   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   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   MonoAP cvdwm 15670   PolyAP cvdwp 15671

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-rmo 2920  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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-rp 11833  df-fz 12327  df-hash 13118  df-vdwap 15672  df-vdwmc 15673  df-vdwpc 15674

This theorem is referenced by:  vdwlem9  15693