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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 12563 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐾 + 1) ∈ ℕ |
10 | 6, 9 | eqeltrri 2823 | . . . . . . 7 ⊢ 𝐼 ∈ ℕ |
11 | nnge1 12292 | . . . . . . 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 469 | . . . . 5 ⊢ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁) |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁)) |
16 | nnz 12631 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
17 | 10, 16 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
19 | 1z 12644 | . . . . . 6 ⊢ 1 ∈ ℤ | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) |
21 | 2 | nnzd 12637 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
22 | elfz 13544 | . . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) | |
23 | 18, 20, 21, 22 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) |
24 | 15, 23 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
25 | lmatfvlem.6 | . . . . . . . 8 ⊢ (𝐿 + 1) = 𝐽 | |
26 | lmatfvlem.2 | . . . . . . . . 9 ⊢ 𝐿 ∈ ℕ0 | |
27 | nn0p1nn 12563 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐿 + 1) ∈ ℕ |
29 | 25, 28 | eqeltrri 2823 | . . . . . . 7 ⊢ 𝐽 ∈ ℕ |
30 | nnge1 12292 | . . . . . . 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 469 | . . . . 5 ⊢ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁) |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁)) |
35 | nnz 12631 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 𝐽 ∈ ℤ) | |
36 | 29, 35 | ax-mp 5 | . . . . . 6 ⊢ 𝐽 ∈ ℤ |
37 | 36 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
38 | elfz 13544 | . . . . 5 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) | |
39 | 37, 20, 21, 38 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) |
40 | 34, 39 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
41 | 1, 2, 3, 4, 5, 24, 40 | lmatfval 33629 | . 2 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) |
42 | 7 | nn0cni 12536 | . . . . . . . 8 ⊢ 𝐾 ∈ ℂ |
43 | ax-1cn 11216 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
44 | 42, 43 | pncan3oi 11526 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = 𝐾 |
45 | 6 | oveq1i 7434 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = (𝐼 − 1) |
46 | 44, 45 | eqtr3i 2756 | . . . . . 6 ⊢ 𝐾 = (𝐼 − 1) |
47 | 46 | fveq2i 6904 | . . . . 5 ⊢ (𝑊‘𝐾) = (𝑊‘(𝐼 − 1)) |
48 | lmatfvlem.7 | . . . . 5 ⊢ (𝑊‘𝐾) = 𝑋 | |
49 | 47, 48 | eqtr3i 2756 | . . . 4 ⊢ (𝑊‘(𝐼 − 1)) = 𝑋 |
50 | 49 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) = 𝑋) |
51 | 50 | fveq1d 6903 | . 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) = (𝑋‘(𝐽 − 1))) |
52 | 26 | nn0cni 12536 | . . . . . . 7 ⊢ 𝐿 ∈ ℂ |
53 | 52, 43 | pncan3oi 11526 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = 𝐿 |
54 | 25 | oveq1i 7434 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = (𝐽 − 1) |
55 | 53, 54 | eqtr3i 2756 | . . . . 5 ⊢ 𝐿 = (𝐽 − 1) |
56 | 55 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐿 = (𝐽 − 1)) |
57 | 56 | fveq2d 6905 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = (𝑋‘(𝐽 − 1))) |
58 | lmatfvlem.8 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) | |
59 | 57, 58 | eqtr3d 2768 | . 2 ⊢ (𝜑 → (𝑋‘(𝐽 − 1)) = 𝑌) |
60 | 41, 51, 59 | 3eqtrd 2770 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 + caddc 11161 ≤ cle 11299 − cmin 11494 ℕcn 12264 ℕ0cn0 12524 ℤcz 12610 ...cfz 13538 ..^cfzo 13681 ♯chash 14347 Word cword 14522 litMatclmat 33626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-lmat 33627 |
This theorem is referenced by: lmat22e12 33634 lmat22e21 33635 lmat22e22 33636 |
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