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 11332 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐾 + 1) ∈ ℕ |
10 | 6, 9 | eqeltrri 2698 | . . . . . . 7 ⊢ 𝐼 ∈ ℕ |
11 | nnge1 11046 | . . . . . . 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 11399 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
17 | 10, 16 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
19 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) |
21 | 2 | nnzd 11481 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
22 | elfz 12332 | . . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) | |
23 | 18, 20, 21, 22 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) |
24 | 15, 23 | mpbird 247 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
25 | lmatfvlem.6 | . . . . . . . 8 ⊢ (𝐿 + 1) = 𝐽 | |
26 | lmatfvlem.2 | . . . . . . . . 9 ⊢ 𝐿 ∈ ℕ0 | |
27 | nn0p1nn 11332 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐿 + 1) ∈ ℕ |
29 | 25, 28 | eqeltrri 2698 | . . . . . . 7 ⊢ 𝐽 ∈ ℕ |
30 | nnge1 11046 | . . . . . . 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 11399 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 𝐽 ∈ ℤ) | |
36 | 29, 35 | ax-mp 5 | . . . . . 6 ⊢ 𝐽 ∈ ℤ |
37 | 36 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
38 | elfz 12332 | . . . . 5 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) | |
39 | 37, 20, 21, 38 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) |
40 | 34, 39 | mpbird 247 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
41 | 1, 2, 3, 4, 5, 24, 40 | lmatfval 29880 | . 2 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) |
42 | 7 | nn0cni 11304 | . . . . . . . 8 ⊢ 𝐾 ∈ ℂ |
43 | ax-1cn 9994 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
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 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) = 𝑋) |
51 | 50 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) = (𝑋‘(𝐽 − 1))) |
52 | 26 | nn0cni 11304 | . . . . . . 7 ⊢ 𝐿 ∈ ℂ |
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 ⊢ (𝜑 → 𝐿 = (𝐽 − 1)) |
57 | 56 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = (𝑋‘(𝐽 − 1))) |
58 | lmatfvlem.8 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) | |
59 | 57, 58 | eqtr3d 2658 | . 2 ⊢ (𝜑 → (𝑋‘(𝐽 − 1)) = 𝑌) |
60 | 41, 51, 59 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ0cn0 11292 ℤcz 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 |
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