Theorem lmatcl 29882

Description: Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)

Hypotheses

Ref	Expression
lmatfval.m	|- M = (litMat ` W )
lmatfval.n	|- ( ph -> N e. NN )
lmatfval.w	|- ( ph -> W e. Word Word V )
lmatfval.1	|- ( ph -> ( # ` W ) = N )
lmatfval.2	|- ( ( ph /\ i e. ( 0..^ N ) ) -> ( # ` ( W ` i ) ) = N )
lmatcl.b	|- V = ( Base ` R )
lmatcl.1	|- O = ( ( 1 ... N ) Mat R )
lmatcl.2	|- P = ( Base ` O )
lmatcl.r	|- ( ph -> R e. X )

Assertion

Ref	Expression
lmatcl	|- ( ph -> M e. P )

Distinct variable groups:   i, M   i, N   i, W   ph, i

Allowed substitution hints:   P( i)   R( i)   O( i)   V( i)   X( i)

Proof of Theorem lmatcl

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

Step	Hyp	Ref	Expression
1		lmatfval.m	. . . 4 |- M = (litMat ` W )
2		lmatfval.w	. . . . 5 |- ( ph -> W e. Word Word V )
3		lmatval 29879	. . . . 5 |- ( W e. Word Word V -> (litMat ` W ) = ( k e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) )
4	2, 3	syl 17	. . . 4 |- ( ph -> (litMat ` W ) = ( k e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) )
5	1, 4	syl5eq 2668	. . 3 |- ( ph -> M = ( k e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) )
6		lmatfval.1	. . . . 5 |- ( ph -> ( # ` W ) = N )
7	6	oveq2d 6666	. . . 4 |- ( ph -> ( 1 ... ( # ` W ) ) = ( 1 ... N ) )
8		lmatfval.n	. . . . . . 7 |- ( ph -> N e. NN )
9		lbfzo0 12507	. . . . . . 7 |- ( 0 e. ( 0..^ N ) <-> N e. NN )
10	8, 9	sylibr 224	. . . . . 6 |- ( ph -> 0 e. ( 0..^ N ) )
11		0nn0 11307	. . . . . . . 8 |- 0 e. NN0
12	11	a1i 11	. . . . . . 7 |- ( ph -> 0 e. NN0 )
13		simpr 477	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> i = 0 )
14	13	eleq1d 2686	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> ( i e. ( 0..^ N ) <-> 0 e. ( 0..^ N ) ) )
15	13	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = 0 ) -> ( W ` i ) = ( W ` 0 ) )
16	15	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( # ` ( W ` i ) ) = ( # ` ( W ` 0 ) ) )
17	16	eqeq1d 2624	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> ( ( # ` ( W ` i ) ) = N <-> ( # ` ( W ` 0 ) ) = N ) )
18	14, 17	imbi12d 334	. . . . . . 7 |- ( ( ph /\ i = 0 ) -> ( ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) <-> ( 0 e. ( 0..^ N ) -> ( # ` ( W ` 0 ) ) = N ) ) )
19		lmatfval.2	. . . . . . . 8 |- ( ( ph /\ i e. ( 0..^ N ) ) -> ( # ` ( W ` i ) ) = N )
20	19	ex 450	. . . . . . 7 |- ( ph -> ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) )
21	12, 18, 20	vtocld 3257	. . . . . 6 |- ( ph -> ( 0 e. ( 0..^ N ) -> ( # ` ( W ` 0 ) ) = N ) )
22	10, 21	mpd 15	. . . . 5 |- ( ph -> ( # ` ( W ` 0 ) ) = N )
23	22	oveq2d 6666	. . . 4 |- ( ph -> ( 1 ... ( # ` ( W ` 0 ) ) ) = ( 1 ... N ) )
24		eqidd 2623	. . . 4 |- ( ph -> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) = ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) )
25	7, 23, 24	mpt2eq123dv 6717	. . 3 |- ( ph -> ( k e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) = ( k e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) )
26	5, 25	eqtrd 2656	. 2 |- ( ph -> M = ( k e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) )
27		lmatcl.1	. . 3 |- O = ( ( 1 ... N ) Mat R )
28		lmatcl.b	. . 3 |- V = ( Base ` R )
29		lmatcl.2	. . 3 |- P = ( Base ` O )
30		fzfid 12772	. . 3 |- ( ph -> ( 1 ... N ) e. Fin )
31		lmatcl.r	. . 3 |- ( ph -> R e. X )
32	2	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> W e. Word Word V )
33		simp2 1062	. . . . . . 7 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> k e. ( 1 ... N ) )
34		fz1fzo0m1 12515	. . . . . . 7 |- ( k e. ( 1 ... N ) -> ( k - 1 ) e. ( 0..^ N ) )
35	33, 34	syl 17	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( k - 1 ) e. ( 0..^ N ) )
36	6	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( # ` W ) = N )
37	36	oveq2d 6666	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( 0..^ ( # ` W ) ) = ( 0..^ N ) )
38	35, 37	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( k - 1 ) e. ( 0..^ ( # ` W ) ) )
39		wrdsymbcl 13318	. . . . 5 |- ( ( W e. Word Word V /\ ( k - 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W ` ( k - 1 ) ) e. Word V )
40	32, 38, 39	syl2anc 693	. . . 4 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( W ` ( k - 1 ) ) e. Word V )
41		simp3 1063	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> j e. ( 1 ... N ) )
42		fz1fzo0m1 12515	. . . . . 6 |- ( j e. ( 1 ... N ) -> ( j - 1 ) e. ( 0..^ N ) )
43	41, 42	syl 17	. . . . 5 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( j - 1 ) e. ( 0..^ N ) )
44		ovexd 6680	. . . . . . . . . 10 |- ( ph -> ( k - 1 ) e. _V )
45		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ i = ( k - 1 ) ) -> i = ( k - 1 ) )
46		eqidd 2623	. . . . . . . . . . . 12 |- ( ( ph /\ i = ( k - 1 ) ) -> ( 0..^ N ) = ( 0..^ N ) )
47	45, 46	eleq12d 2695	. . . . . . . . . . 11 |- ( ( ph /\ i = ( k - 1 ) ) -> ( i e. ( 0..^ N ) <-> ( k - 1 ) e. ( 0..^ N ) ) )
48	45	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ph /\ i = ( k - 1 ) ) -> ( W ` i ) = ( W ` ( k - 1 ) ) )
49	48	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ i = ( k - 1 ) ) -> ( # ` ( W ` i ) ) = ( # ` ( W ` ( k - 1 ) ) ) )
50	49	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( ph /\ i = ( k - 1 ) ) -> ( ( # ` ( W ` i ) ) = N <-> ( # ` ( W ` ( k - 1 ) ) ) = N ) )
51	47, 50	imbi12d 334	. . . . . . . . . 10 |- ( ( ph /\ i = ( k - 1 ) ) -> ( ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) <-> ( ( k - 1 ) e. ( 0..^ N ) -> ( # ` ( W ` ( k - 1 ) ) ) = N ) ) )
52	44, 51, 20	vtocld 3257	. . . . . . . . 9 |- ( ph -> ( ( k - 1 ) e. ( 0..^ N ) -> ( # ` ( W ` ( k - 1 ) ) ) = N ) )
53	52	imp 445	. . . . . . . 8 |- ( ( ph /\ ( k - 1 ) e. ( 0..^ N ) ) -> ( # ` ( W ` ( k - 1 ) ) ) = N )
54	34, 53	sylan2 491	. . . . . . 7 |- ( ( ph /\ k e. ( 1 ... N ) ) -> ( # ` ( W ` ( k - 1 ) ) ) = N )
55	54	3adant3 1081	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( # ` ( W ` ( k - 1 ) ) ) = N )
56	55	oveq2d 6666	. . . . 5 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( 0..^ ( # ` ( W ` ( k - 1 ) ) ) ) = ( 0..^ N ) )
57	43, 56	eleqtrrd 2704	. . . 4 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( j - 1 ) e. ( 0..^ ( # ` ( W ` ( k - 1 ) ) ) ) )
58		wrdsymbcl 13318	. . . 4 |- ( ( ( W ` ( k - 1 ) ) e. Word V /\ ( j - 1 ) e. ( 0..^ ( # ` ( W ` ( k - 1 ) ) ) ) ) -> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) e. V )
59	40, 57, 58	syl2anc 693	. . 3 |- ( ( ph /\ k e. ( 1 ... N ) /\ j e. ( 1 ... N ) ) -> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) e. V )
60	27, 28, 29, 30, 31, 59	matbas2d 20229	. 2 |- ( ph -> ( k e. ( 1 ... N ) , j e. ( 1 ... N ) |-> ( ( W ` ( k - 1 ) ) ` ( j - 1 ) ) ) e. P )
61	26, 60	eqeltrd 2701	1 |- ( ph -> M e. P )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   Basecbs 15857   Mat cmat 20213  litMatclmat 29877

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-ot 4186  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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-card 8765  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-lmat 29878

This theorem is referenced by:  lmat22det  29888