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 11209 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐾 + 1) ∈ ℕ |
10 | 6, 9 | eqeltrri 2685 | . . . . . . 7 ⊢ 𝐼 ∈ ℕ |
11 | nnge1 10923 | . . . . . . 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 470 | . . . . 5 ⊢ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁) |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁)) |
16 | nnz 11276 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
17 | 10, 16 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
19 | 1z 11284 | . . . . . 6 ⊢ 1 ∈ ℤ | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) |
21 | 2 | nnzd 11357 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
22 | elfz 12203 | . . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) | |
23 | 18, 20, 21, 22 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) |
24 | 15, 23 | mpbird 246 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
25 | lmatfvlem.6 | . . . . . . . 8 ⊢ (𝐿 + 1) = 𝐽 | |
26 | lmatfvlem.2 | . . . . . . . . 9 ⊢ 𝐿 ∈ ℕ0 | |
27 | nn0p1nn 11209 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐿 + 1) ∈ ℕ |
29 | 25, 28 | eqeltrri 2685 | . . . . . . 7 ⊢ 𝐽 ∈ ℕ |
30 | nnge1 10923 | . . . . . . 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 470 | . . . . 5 ⊢ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁) |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁)) |
35 | nnz 11276 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 𝐽 ∈ ℤ) | |
36 | 29, 35 | ax-mp 5 | . . . . . 6 ⊢ 𝐽 ∈ ℤ |
37 | 36 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
38 | elfz 12203 | . . . . 5 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) | |
39 | 37, 20, 21, 38 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) |
40 | 34, 39 | mpbird 246 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
41 | 1, 2, 3, 4, 5, 24, 40 | lmatfval 29208 | . 2 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) |
42 | 7 | nn0cni 11181 | . . . . . . . 8 ⊢ 𝐾 ∈ ℂ |
43 | ax-1cn 9873 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
44 | 42, 43 | pncan3oi 10176 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = 𝐾 |
45 | 6 | oveq1i 6559 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = (𝐼 − 1) |
46 | 44, 45 | eqtr3i 2634 | . . . . . 6 ⊢ 𝐾 = (𝐼 − 1) |
47 | 46 | fveq2i 6106 | . . . . 5 ⊢ (𝑊‘𝐾) = (𝑊‘(𝐼 − 1)) |
48 | lmatfvlem.7 | . . . . 5 ⊢ (𝑊‘𝐾) = 𝑋 | |
49 | 47, 48 | eqtr3i 2634 | . . . 4 ⊢ (𝑊‘(𝐼 − 1)) = 𝑋 |
50 | 49 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) = 𝑋) |
51 | 50 | fveq1d 6105 | . 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) = (𝑋‘(𝐽 − 1))) |
52 | 26 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐿 ∈ ℂ |
53 | 52, 43 | pncan3oi 10176 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = 𝐿 |
54 | 25 | oveq1i 6559 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = (𝐽 − 1) |
55 | 53, 54 | eqtr3i 2634 | . . . . 5 ⊢ 𝐿 = (𝐽 − 1) |
56 | 55 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐿 = (𝐽 − 1)) |
57 | 56 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = (𝑋‘(𝐽 − 1))) |
58 | lmatfvlem.8 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) | |
59 | 57, 58 | eqtr3d 2646 | . 2 ⊢ (𝜑 → (𝑋‘(𝐽 − 1)) = 𝑌) |
60 | 41, 51, 59 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 litMatclmat 29205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lmat 29206 |
This theorem is referenced by: lmat22e12 29213 lmat22e21 29214 lmat22e22 29215 |
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