lmatfval - Mathbox for Thierry Arnoux

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Theorem lmatfval 33324

Description: Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)

**Hypotheses**
Ref	Expression
lmatfval.m	⊢ 𝑀 = (litMat‘𝑊)
lmatfval.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
lmatfval.w	⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)
lmatfval.1	⊢ (𝜑 → (♯‘𝑊) = 𝑁)
lmatfval.2	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)
lmatfval.i	⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
lmatfval.j	⊢ (𝜑 → 𝐽 ∈ (1...𝑁))

**Assertion**
Ref	Expression
lmatfval	⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))

Distinct variable groups: 𝑖,𝑀 𝑖,𝐼 𝑖,𝐽 𝑖,𝑁 𝑖,𝑊 𝜑,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 ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)
3		lmatval 33323	. . . 4 ⊢ (𝑊 ∈ Word Word 𝑉 → (litMat‘𝑊) = (𝑖 ∈ (1...(♯‘𝑊)), 𝑗 ∈ (1...(♯‘(𝑊‘0))) ↦ ((𝑊‘(𝑖 − 1))‘(𝑗 − 1))))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (litMat‘𝑊) = (𝑖 ∈ (1...(♯‘𝑊)), 𝑗 ∈ (1...(♯‘(𝑊‘0))) ↦ ((𝑊‘(𝑖 − 1))‘(𝑗 − 1))))
5	1, 4	eqtrid 2778	. 2 ⊢ (𝜑 → 𝑀 = (𝑖 ∈ (1...(♯‘𝑊)), 𝑗 ∈ (1...(♯‘(𝑊‘0))) ↦ ((𝑊‘(𝑖 − 1))‘(𝑗 − 1))))
6		simprl 768	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → 𝑖 = 𝐼)
7	6	fvoveq1d 7427	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → (𝑊‘(𝑖 − 1)) = (𝑊‘(𝐼 − 1)))
8		simprr 770	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → 𝑗 = 𝐽)
9	8	oveq1d 7420	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → (𝑗 − 1) = (𝐽 − 1))
10	7, 9	fveq12d 6892	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝐽)) → ((𝑊‘(𝑖 − 1))‘(𝑗 − 1)) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
11		lmatfval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
12		lmatfval.1	. . . 4 ⊢ (𝜑 → (♯‘𝑊) = 𝑁)
13	12	oveq2d 7421	. . 3 ⊢ (𝜑 → (1...(♯‘𝑊)) = (1...𝑁))
14	11, 13	eleqtrrd 2830	. 2 ⊢ (𝜑 → 𝐼 ∈ (1...(♯‘𝑊)))
15		lmatfval.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
16		1m1e0 12288	. . . . . 6 ⊢ (1 − 1) = 0
17		lmatfval.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18		nnuz 12869	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	eleqtrdi 2837	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
20		eluzfz1 13514	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (1...𝑁))
22		fz1fzo0m1 13686	. . . . . . 7 ⊢ (1 ∈ (1...𝑁) → (1 − 1) ∈ (0..^𝑁))
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (1 − 1) ∈ (0..^𝑁))
24	16, 23	eqeltrrid 2832	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^𝑁))
25		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0)
26	25	eleq1d 2812	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑖 ∈ (0..^𝑁) ↔ 0 ∈ (0..^𝑁)))
27	25	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑊‘𝑖) = (𝑊‘0))
28	27	fveqeq2d 6893	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((♯‘(𝑊‘𝑖)) = 𝑁 ↔ (♯‘(𝑊‘0)) = 𝑁))
29	26, 28	imbi12d 344	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑖 ∈ (0..^𝑁) → (♯‘(𝑊‘𝑖)) = 𝑁) ↔ (0 ∈ (0..^𝑁) → (♯‘(𝑊‘0)) = 𝑁)))
30		lmatfval.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)
31	30	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑁) → (♯‘(𝑊‘𝑖)) = 𝑁))
32	24, 29, 31	vtocld 3542	. . . . 5 ⊢ (𝜑 → (0 ∈ (0..^𝑁) → (♯‘(𝑊‘0)) = 𝑁))
33	24, 32	mpd 15	. . . 4 ⊢ (𝜑 → (♯‘(𝑊‘0)) = 𝑁)
34	33	oveq2d 7421	. . 3 ⊢ (𝜑 → (1...(♯‘(𝑊‘0))) = (1...𝑁))
35	15, 34	eleqtrrd 2830	. 2 ⊢ (𝜑 → 𝐽 ∈ (1...(♯‘(𝑊‘0))))
36		fz1fzo0m1 13686	. . . . . 6 ⊢ (𝐼 ∈ (1...𝑁) → (𝐼 − 1) ∈ (0..^𝑁))
37	11, 36	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 − 1) ∈ (0..^𝑁))
38	12	oveq2d 7421	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝑁))
39	37, 38	eleqtrrd 2830	. . . 4 ⊢ (𝜑 → (𝐼 − 1) ∈ (0..^(♯‘𝑊)))
40		wrdsymbcl 14483	. . . 4 ⊢ ((𝑊 ∈ Word Word 𝑉 ∧ (𝐼 − 1) ∈ (0..^(♯‘𝑊))) → (𝑊‘(𝐼 − 1)) ∈ Word 𝑉)
41	2, 39, 40	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) ∈ Word 𝑉)
42		fz1fzo0m1 13686	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → (𝐽 − 1) ∈ (0..^𝑁))
43	15, 42	syl 17	. . . 4 ⊢ (𝜑 → (𝐽 − 1) ∈ (0..^𝑁))
44		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = (𝐼 − 1)) → 𝑖 = (𝐼 − 1))
45	44	eleq1d 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = (𝐼 − 1)) → (𝑖 ∈ (0..^𝑁) ↔ (𝐼 − 1) ∈ (0..^𝑁)))
46	44	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = (𝐼 − 1)) → (𝑊‘𝑖) = (𝑊‘(𝐼 − 1)))
47	46	fveqeq2d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = (𝐼 − 1)) → ((♯‘(𝑊‘𝑖)) = 𝑁 ↔ (♯‘(𝑊‘(𝐼 − 1))) = 𝑁))
48	45, 47	imbi12d 344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = (𝐼 − 1)) → ((𝑖 ∈ (0..^𝑁) → (♯‘(𝑊‘𝑖)) = 𝑁) ↔ ((𝐼 − 1) ∈ (0..^𝑁) → (♯‘(𝑊‘(𝐼 − 1))) = 𝑁)))
49	37, 48, 31	vtocld 3542	. . . . . 6 ⊢ (𝜑 → ((𝐼 − 1) ∈ (0..^𝑁) → (♯‘(𝑊‘(𝐼 − 1))) = 𝑁))
50	37, 49	mpd 15	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊‘(𝐼 − 1))) = 𝑁)
51	50	oveq2d 7421	. . . 4 ⊢ (𝜑 → (0..^(♯‘(𝑊‘(𝐼 − 1)))) = (0..^𝑁))
52	43, 51	eleqtrrd 2830	. . 3 ⊢ (𝜑 → (𝐽 − 1) ∈ (0..^(♯‘(𝑊‘(𝐼 − 1)))))
53		wrdsymbcl 14483	. . 3 ⊢ (((𝑊‘(𝐼 − 1)) ∈ Word 𝑉 ∧ (𝐽 − 1) ∈ (0..^(♯‘(𝑊‘(𝐼 − 1))))) → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) ∈ 𝑉)
54	41, 52, 53	syl2anc 583	. 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) ∈ 𝑉)
55	5, 10, 14, 35, 54	ovmpod 7556	1 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 0cc0 11112 1c1 11113 − cmin 11448 ℕcn 12216 ℤ_≥cuz 12826 ...cfz 13490 ..^cfzo 13633 ♯chash 14295 Word cword 14470 litMatclmat 33321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-lmat 33322

This theorem is referenced by: lmatfvlem 33325 lmat22e11 33328

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