Theorem lmatfvlem 29881

Description: Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-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 )
lmatfvlem.1	|- K e. NN0
lmatfvlem.2	|- L e. NN0
lmatfvlem.3	|- I <_ N
lmatfvlem.4	|- J <_ N
lmatfvlem.5	|- ( K + 1 ) = I
lmatfvlem.6	|- ( L + 1 ) = J
lmatfvlem.7	|- ( W ` K ) = X
lmatfvlem.8	|- ( ph -> ( X ` L ) = Y )

Assertion

Ref	Expression
lmatfvlem	|- ( ph -> ( I M J ) = Y )

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

Allowed substitution hints:   K( i)   L( i)   V( i)   X( i)   Y( i)

Proof of Theorem lmatfvlem

Step	Hyp	Ref	Expression
1		lmatfval.m	. . 3 |- M = (litMat ` W )
2		lmatfval.n	. . 3 |- ( ph -> N e. NN )
3		lmatfval.w	. . 3 |- ( ph -> W e. Word Word V )
4		lmatfval.1	. . 3 |- ( ph -> ( # ` W ) = N )
5		lmatfval.2	. . 3 |- ( ( ph /\ i e. ( 0..^ N ) ) -> ( # ` ( W ` i ) ) = N )
6		lmatfvlem.5	. . . . . . . 8 |- ( K + 1 ) = I
7		lmatfvlem.1	. . . . . . . . 9 |- K e. NN0
8		nn0p1nn 11332	. . . . . . . . 9 |- ( K e. NN0 -> ( K + 1 ) e. NN )
9	7, 8	ax-mp 5	. . . . . . . 8 |- ( K + 1 ) e. NN
10	6, 9	eqeltrri 2698	. . . . . . 7 |- I e. NN
11		nnge1 11046	. . . . . . 7 |- ( I e. NN -> 1 <_ I )
12	10, 11	ax-mp 5	. . . . . 6 |- 1 <_ I
13		lmatfvlem.3	. . . . . 6 |- I <_ N
14	12, 13	pm3.2i 471	. . . . 5 |- ( 1 <_ I /\ I <_ N )
15	14	a1i 11	. . . 4 |- ( ph -> ( 1 <_ I /\ I <_ N ) )
16		nnz 11399	. . . . . . 7 |- ( I e. NN -> I e. ZZ )
17	10, 16	ax-mp 5	. . . . . 6 |- I e. ZZ
18	17	a1i 11	. . . . 5 |- ( ph -> I e. ZZ )
19		1z 11407	. . . . . 6 |- 1 e. ZZ
20	19	a1i 11	. . . . 5 |- ( ph -> 1 e. ZZ )
21	2	nnzd 11481	. . . . 5 |- ( ph -> N e. ZZ )
22		elfz 12332	. . . . 5 |- ( ( I e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) -> ( I e. ( 1 ... N ) <-> ( 1 <_ I /\ I <_ N ) ) )
23	18, 20, 21, 22	syl3anc 1326	. . . 4 |- ( ph -> ( I e. ( 1 ... N ) <-> ( 1 <_ I /\ I <_ N ) ) )
24	15, 23	mpbird 247	. . 3 |- ( ph -> I e. ( 1 ... N ) )
25		lmatfvlem.6	. . . . . . . 8 |- ( L + 1 ) = J
26		lmatfvlem.2	. . . . . . . . 9 |- L e. NN0
27		nn0p1nn 11332	. . . . . . . . 9 |- ( L e. NN0 -> ( L + 1 ) e. NN )
28	26, 27	ax-mp 5	. . . . . . . 8 |- ( L + 1 ) e. NN
29	25, 28	eqeltrri 2698	. . . . . . 7 |- J e. NN
30		nnge1 11046	. . . . . . 7 |- ( J e. NN -> 1 <_ J )
31	29, 30	ax-mp 5	. . . . . 6 |- 1 <_ J
32		lmatfvlem.4	. . . . . 6 |- J <_ N
33	31, 32	pm3.2i 471	. . . . 5 |- ( 1 <_ J /\ J <_ N )
34	33	a1i 11	. . . 4 |- ( ph -> ( 1 <_ J /\ J <_ N ) )
35		nnz 11399	. . . . . . 7 |- ( J e. NN -> J e. ZZ )
36	29, 35	ax-mp 5	. . . . . 6 |- J e. ZZ
37	36	a1i 11	. . . . 5 |- ( ph -> J e. ZZ )
38		elfz 12332	. . . . 5 |- ( ( J e. ZZ /\ 1 e. ZZ /\ N e. ZZ ) -> ( J e. ( 1 ... N ) <-> ( 1 <_ J /\ J <_ N ) ) )
39	37, 20, 21, 38	syl3anc 1326	. . . 4 |- ( ph -> ( J e. ( 1 ... N ) <-> ( 1 <_ J /\ J <_ N ) ) )
40	34, 39	mpbird 247	. . 3 |- ( ph -> J e. ( 1 ... N ) )
41	1, 2, 3, 4, 5, 24, 40	lmatfval 29880	. 2 |- ( ph -> ( I M J ) = ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) )
42	7	nn0cni 11304	. . . . . . . 8 |- K e. CC
43		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
44	42, 43	pncan3oi 10297	. . . . . . 7 |- ( ( K + 1 ) - 1 ) = K
45	6	oveq1i 6660	. . . . . . 7 |- ( ( K + 1 ) - 1 ) = ( I - 1 )
46	44, 45	eqtr3i 2646	. . . . . 6 |- K = ( I - 1 )
47	46	fveq2i 6194	. . . . 5 |- ( W ` K ) = ( W ` ( I - 1 ) )
48		lmatfvlem.7	. . . . 5 |- ( W ` K ) = X
49	47, 48	eqtr3i 2646	. . . 4 |- ( W ` ( I - 1 ) ) = X
50	49	a1i 11	. . 3 |- ( ph -> ( W ` ( I - 1 ) ) = X )
51	50	fveq1d 6193	. 2 |- ( ph -> ( ( W ` ( I - 1 ) ) ` ( J - 1 ) ) = ( X ` ( J - 1 ) ) )
52	26	nn0cni 11304	. . . . . . 7 |- L e. CC
53	52, 43	pncan3oi 10297	. . . . . 6 |- ( ( L + 1 ) - 1 ) = L
54	25	oveq1i 6660	. . . . . 6 |- ( ( L + 1 ) - 1 ) = ( J - 1 )
55	53, 54	eqtr3i 2646	. . . . 5 |- L = ( J - 1 )
56	55	a1i 11	. . . 4 |- ( ph -> L = ( J - 1 ) )
57	56	fveq2d 6195	. . 3 |- ( ph -> ( X ` L ) = ( X ` ( J - 1 ) ) )
58		lmatfvlem.8	. . 3 |- ( ph -> ( X ` L ) = Y )
59	57, 58	eqtr3d 2658	. 2 |- ( ph -> ( X ` ( J - 1 ) ) = Y )
60	41, 51, 59	3eqtrd 2660	1 |- ( ph -> ( I M J ) = Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...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:  lmat22e12  29885  lmat22e21  29886  lmat22e22  29887