Theorem lmatfval 29880

Description: Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-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 )
lmatfval.i	|- ( ph -> I e. ( 1 ... N ) )
lmatfval.j	|- ( ph -> J e. ( 1 ... N ) )

Assertion

Ref	Expression
lmatfval	|- ( ph -> ( I M J ) = ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) )

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

Allowed substitution hint:   V( i)

Proof of Theorem lmatfval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmatfval.m	. . 3 |- M = (litMat ` W )
2		lmatfval.w	. . . 4 |- ( ph -> W e. Word Word V )
3		lmatval 29879	. . . 4 |- ( W e. Word Word V -> (litMat ` W ) = ( i e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( i - 1 ) ) ` ( j - 1 ) ) ) )
4	2, 3	syl 17	. . 3 |- ( ph -> (litMat ` W ) = ( i e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( i - 1 ) ) ` ( j - 1 ) ) ) )
5	1, 4	syl5eq 2668	. 2 |- ( ph -> M = ( i e. ( 1 ... ( # ` W ) ) , j e. ( 1 ... ( # ` ( W ` 0 ) ) ) |-> ( ( W ` ( i - 1 ) ) ` ( j - 1 ) ) ) )
6		simprl 794	. . . . 5 |- ( ( ph /\ ( i = I /\ j = J ) ) -> i = I )
7	6	oveq1d 6665	. . . 4 |- ( ( ph /\ ( i = I /\ j = J ) ) -> ( i - 1 ) = ( I - 1 ) )
8	7	fveq2d 6195	. . 3 |- ( ( ph /\ ( i = I /\ j = J ) ) -> ( W ` ( i - 1 ) ) = ( W ` ( I - 1 ) ) )
9		simprr 796	. . . 4 |- ( ( ph /\ ( i = I /\ j = J ) ) -> j = J )
10	9	oveq1d 6665	. . 3 |- ( ( ph /\ ( i = I /\ j = J ) ) -> ( j - 1 ) = ( J - 1 ) )
11	8, 10	fveq12d 6197	. 2 |- ( ( ph /\ ( i = I /\ j = J ) ) -> ( ( W ` ( i - 1 ) ) ` ( j - 1 ) ) = ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) )
12		lmatfval.i	. . 3 |- ( ph -> I e. ( 1 ... N ) )
13		lmatfval.1	. . . 4 |- ( ph -> ( # ` W ) = N )
14	13	oveq2d 6666	. . 3 |- ( ph -> ( 1 ... ( # ` W ) ) = ( 1 ... N ) )
15	12, 14	eleqtrrd 2704	. 2 |- ( ph -> I e. ( 1 ... ( # ` W ) ) )
16		lmatfval.j	. . 3 |- ( ph -> J e. ( 1 ... N ) )
17		1m1e0 11089	. . . . . 6 |- ( 1 - 1 ) = 0
18		lmatfval.n	. . . . . . . . 9 |- ( ph -> N e. NN )
19		nnuz 11723	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
20	18, 19	syl6eleq 2711	. . . . . . . 8 |- ( ph -> N e. ( ZZ>= ` 1 ) )
21		eluzfz1 12348	. . . . . . . 8 |- ( N e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... N ) )
22	20, 21	syl 17	. . . . . . 7 |- ( ph -> 1 e. ( 1 ... N ) )
23		fz1fzo0m1 12515	. . . . . . 7 |- ( 1 e. ( 1 ... N ) -> ( 1 - 1 ) e. ( 0..^ N ) )
24	22, 23	syl 17	. . . . . 6 |- ( ph -> ( 1 - 1 ) e. ( 0..^ N ) )
25	17, 24	syl5eqelr 2706	. . . . 5 |- ( ph -> 0 e. ( 0..^ N ) )
26		simpr 477	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> i = 0 )
27	26	eleq1d 2686	. . . . . . 7 |- ( ( ph /\ i = 0 ) -> ( i e. ( 0..^ N ) <-> 0 e. ( 0..^ N ) ) )
28	26	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ i = 0 ) -> ( W ` i ) = ( W ` 0 ) )
29	28	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ i = 0 ) -> ( # ` ( W ` i ) ) = ( # ` ( W ` 0 ) ) )
30	29	eqeq1d 2624	. . . . . . 7 |- ( ( ph /\ i = 0 ) -> ( ( # ` ( W ` i ) ) = N <-> ( # ` ( W ` 0 ) ) = N ) )
31	27, 30	imbi12d 334	. . . . . 6 |- ( ( ph /\ i = 0 ) -> ( ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) <-> ( 0 e. ( 0..^ N ) -> ( # ` ( W ` 0 ) ) = N ) ) )
32		lmatfval.2	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ N ) ) -> ( # ` ( W ` i ) ) = N )
33	32	ex 450	. . . . . 6 |- ( ph -> ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) )
34	25, 31, 33	vtocld 3257	. . . . 5 |- ( ph -> ( 0 e. ( 0..^ N ) -> ( # ` ( W ` 0 ) ) = N ) )
35	25, 34	mpd 15	. . . 4 |- ( ph -> ( # ` ( W ` 0 ) ) = N )
36	35	oveq2d 6666	. . 3 |- ( ph -> ( 1 ... ( # ` ( W ` 0 ) ) ) = ( 1 ... N ) )
37	16, 36	eleqtrrd 2704	. 2 |- ( ph -> J e. ( 1 ... ( # ` ( W ` 0 ) ) ) )
38		fz1fzo0m1 12515	. . . . . 6 |- ( I e. ( 1 ... N ) -> ( I - 1 ) e. ( 0..^ N ) )
39	12, 38	syl 17	. . . . 5 |- ( ph -> ( I - 1 ) e. ( 0..^ N ) )
40	13	oveq2d 6666	. . . . 5 |- ( ph -> ( 0..^ ( # ` W ) ) = ( 0..^ N ) )
41	39, 40	eleqtrrd 2704	. . . 4 |- ( ph -> ( I - 1 ) e. ( 0..^ ( # ` W ) ) )
42		wrdsymbcl 13318	. . . 4 |- ( ( W e. Word Word V /\ ( I - 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W ` ( I - 1 ) ) e. Word V )
43	2, 41, 42	syl2anc 693	. . 3 |- ( ph -> ( W ` ( I - 1 ) ) e. Word V )
44		fz1fzo0m1 12515	. . . . 5 |- ( J e. ( 1 ... N ) -> ( J - 1 ) e. ( 0..^ N ) )
45	16, 44	syl 17	. . . 4 |- ( ph -> ( J - 1 ) e. ( 0..^ N ) )
46		simpr 477	. . . . . . . . 9 |- ( ( ph /\ i = ( I - 1 ) ) -> i = ( I - 1 ) )
47	46	eleq1d 2686	. . . . . . . 8 |- ( ( ph /\ i = ( I - 1 ) ) -> ( i e. ( 0..^ N ) <-> ( I - 1 ) e. ( 0..^ N ) ) )
48	46	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ i = ( I - 1 ) ) -> ( W ` i ) = ( W ` ( I - 1 ) ) )
49	48	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ i = ( I - 1 ) ) -> ( # ` ( W ` i ) ) = ( # ` ( W ` ( I - 1 ) ) ) )
50	49	eqeq1d 2624	. . . . . . . 8 |- ( ( ph /\ i = ( I - 1 ) ) -> ( ( # ` ( W ` i ) ) = N <-> ( # ` ( W ` ( I - 1 ) ) ) = N ) )
51	47, 50	imbi12d 334	. . . . . . 7 |- ( ( ph /\ i = ( I - 1 ) ) -> ( ( i e. ( 0..^ N ) -> ( # ` ( W ` i ) ) = N ) <-> ( ( I - 1 ) e. ( 0..^ N ) -> ( # ` ( W ` ( I - 1 ) ) ) = N ) ) )
52	39, 51, 33	vtocld 3257	. . . . . 6 |- ( ph -> ( ( I - 1 ) e. ( 0..^ N ) -> ( # ` ( W ` ( I - 1 ) ) ) = N ) )
53	39, 52	mpd 15	. . . . 5 |- ( ph -> ( # ` ( W ` ( I - 1 ) ) ) = N )
54	53	oveq2d 6666	. . . 4 |- ( ph -> ( 0..^ ( # ` ( W ` ( I - 1 ) ) ) ) = ( 0..^ N ) )
55	45, 54	eleqtrrd 2704	. . 3 |- ( ph -> ( J - 1 ) e. ( 0..^ ( # ` ( W ` ( I - 1 ) ) ) ) )
56		wrdsymbcl 13318	. . 3 |- ( ( ( W ` ( I - 1 ) ) e. Word V /\ ( J - 1 ) e. ( 0..^ ( # ` ( W ` ( I - 1 ) ) ) ) ) -> ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) e. V )
57	43, 55, 56	syl2anc 693	. 2 |- ( ph -> ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) e. V )
58	5, 11, 15, 37, 57	ovmpt2d 6788	1 |- ( ph -> ( I M J ) = ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lmat 29878

This theorem is referenced by:  lmatfvlem  29881  lmat22e11  29884