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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmatfvlem | Structured version Visualization version GIF version |
Description: Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.) |
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) |
lmatfvlem.1 | ⊢ 𝐾 ∈ ℕ0 |
lmatfvlem.2 | ⊢ 𝐿 ∈ ℕ0 |
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 | ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) |
Ref | Expression |
---|---|
lmatfvlem | ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmatfval.m | . . 3 ⊢ 𝑀 = (litMat‘𝑊) | |
2 | lmatfval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | lmatfval.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) | |
4 | lmatfval.1 | . . 3 ⊢ (𝜑 → (♯‘𝑊) = 𝑁) | |
5 | lmatfval.2 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) | |
6 | lmatfvlem.5 | . . . . . . . 8 ⊢ (𝐾 + 1) = 𝐼 | |
7 | lmatfvlem.1 | . . . . . . . . 9 ⊢ 𝐾 ∈ ℕ0 | |
8 | nn0p1nn 11773 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐾 + 1) ∈ ℕ |
10 | 6, 9 | eqeltrri 2878 | . . . . . . 7 ⊢ 𝐼 ∈ ℕ |
11 | nnge1 11502 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 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 ⊢ (𝜑 → (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁)) |
16 | nnz 11842 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
17 | 10, 16 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
19 | 1z 11850 | . . . . . 6 ⊢ 1 ∈ ℤ | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) |
21 | 2 | nnzd 11924 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
22 | elfz 12737 | . . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) | |
23 | 18, 20, 21, 22 | syl3anc 1362 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) |
24 | 15, 23 | mpbird 258 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
25 | lmatfvlem.6 | . . . . . . . 8 ⊢ (𝐿 + 1) = 𝐽 | |
26 | lmatfvlem.2 | . . . . . . . . 9 ⊢ 𝐿 ∈ ℕ0 | |
27 | nn0p1nn 11773 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐿 + 1) ∈ ℕ |
29 | 25, 28 | eqeltrri 2878 | . . . . . . 7 ⊢ 𝐽 ∈ ℕ |
30 | nnge1 11502 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 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 ⊢ (𝜑 → (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁)) |
35 | nnz 11842 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 𝐽 ∈ ℤ) | |
36 | 29, 35 | ax-mp 5 | . . . . . 6 ⊢ 𝐽 ∈ ℤ |
37 | 36 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
38 | elfz 12737 | . . . . 5 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) | |
39 | 37, 20, 21, 38 | syl3anc 1362 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) |
40 | 34, 39 | mpbird 258 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
41 | 1, 2, 3, 4, 5, 24, 40 | lmatfval 30650 | . 2 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) |
42 | 7 | nn0cni 11746 | . . . . . . . 8 ⊢ 𝐾 ∈ ℂ |
43 | ax-1cn 10430 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
44 | 42, 43 | pncan3oi 10739 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = 𝐾 |
45 | 6 | oveq1i 7017 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = (𝐼 − 1) |
46 | 44, 45 | eqtr3i 2819 | . . . . . 6 ⊢ 𝐾 = (𝐼 − 1) |
47 | 46 | fveq2i 6533 | . . . . 5 ⊢ (𝑊‘𝐾) = (𝑊‘(𝐼 − 1)) |
48 | lmatfvlem.7 | . . . . 5 ⊢ (𝑊‘𝐾) = 𝑋 | |
49 | 47, 48 | eqtr3i 2819 | . . . 4 ⊢ (𝑊‘(𝐼 − 1)) = 𝑋 |
50 | 49 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) = 𝑋) |
51 | 50 | fveq1d 6532 | . 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) = (𝑋‘(𝐽 − 1))) |
52 | 26 | nn0cni 11746 | . . . . . . 7 ⊢ 𝐿 ∈ ℂ |
53 | 52, 43 | pncan3oi 10739 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = 𝐿 |
54 | 25 | oveq1i 7017 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = (𝐽 − 1) |
55 | 53, 54 | eqtr3i 2819 | . . . . 5 ⊢ 𝐿 = (𝐽 − 1) |
56 | 55 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐿 = (𝐽 − 1)) |
57 | 56 | fveq2d 6534 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = (𝑋‘(𝐽 − 1))) |
58 | lmatfvlem.8 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) | |
59 | 57, 58 | eqtr3d 2831 | . 2 ⊢ (𝜑 → (𝑋‘(𝐽 − 1)) = 𝑌) |
60 | 41, 51, 59 | 3eqtrd 2833 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 class class class wbr 4956 ‘cfv 6217 (class class class)co 7007 0cc0 10372 1c1 10373 + caddc 10375 ≤ cle 10511 − cmin 10706 ℕcn 11475 ℕ0cn0 11734 ℤcz 11818 ...cfz 12731 ..^cfzo 12872 ♯chash 13528 Word cword 13695 litMatclmat 30647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-fzo 12873 df-hash 13529 df-word 13696 df-lmat 30648 |
This theorem is referenced by: lmat22e12 30655 lmat22e21 30656 lmat22e22 30657 |
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