| 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 12565 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ) | |
| 9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐾 + 1) ∈ ℕ |
| 10 | 6, 9 | eqeltrri 2838 | . . . . . . 7 ⊢ 𝐼 ∈ ℕ |
| 11 | nnge1 12294 | . . . . . . 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 12634 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
| 17 | 10, 16 | ax-mp 5 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
| 18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
| 19 | 1z 12647 | . . . . . 6 ⊢ 1 ∈ ℤ | |
| 20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) |
| 21 | 2 | nnzd 12640 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 22 | elfz 13553 | . . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) | |
| 23 | 18, 20, 21, 22 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ (1...𝑁) ↔ (1 ≤ 𝐼 ∧ 𝐼 ≤ 𝑁))) |
| 24 | 15, 23 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
| 25 | lmatfvlem.6 | . . . . . . . 8 ⊢ (𝐿 + 1) = 𝐽 | |
| 26 | lmatfvlem.2 | . . . . . . . . 9 ⊢ 𝐿 ∈ ℕ0 | |
| 27 | nn0p1nn 12565 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ) | |
| 28 | 26, 27 | ax-mp 5 | . . . . . . . 8 ⊢ (𝐿 + 1) ∈ ℕ |
| 29 | 25, 28 | eqeltrri 2838 | . . . . . . 7 ⊢ 𝐽 ∈ ℕ |
| 30 | nnge1 12294 | . . . . . . 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 12634 | . . . . . . 7 ⊢ (𝐽 ∈ ℕ → 𝐽 ∈ ℤ) | |
| 36 | 29, 35 | ax-mp 5 | . . . . . 6 ⊢ 𝐽 ∈ ℤ |
| 37 | 36 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
| 38 | elfz 13553 | . . . . 5 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) | |
| 39 | 37, 20, 21, 38 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (1...𝑁) ↔ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑁))) |
| 40 | 34, 39 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
| 41 | 1, 2, 3, 4, 5, 24, 40 | lmatfval 33813 | . 2 ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) |
| 42 | 7 | nn0cni 12538 | . . . . . . . 8 ⊢ 𝐾 ∈ ℂ |
| 43 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 44 | 42, 43 | pncan3oi 11524 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = 𝐾 |
| 45 | 6 | oveq1i 7441 | . . . . . . 7 ⊢ ((𝐾 + 1) − 1) = (𝐼 − 1) |
| 46 | 44, 45 | eqtr3i 2767 | . . . . . 6 ⊢ 𝐾 = (𝐼 − 1) |
| 47 | 46 | fveq2i 6909 | . . . . 5 ⊢ (𝑊‘𝐾) = (𝑊‘(𝐼 − 1)) |
| 48 | lmatfvlem.7 | . . . . 5 ⊢ (𝑊‘𝐾) = 𝑋 | |
| 49 | 47, 48 | eqtr3i 2767 | . . . 4 ⊢ (𝑊‘(𝐼 − 1)) = 𝑋 |
| 50 | 49 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑊‘(𝐼 − 1)) = 𝑋) |
| 51 | 50 | fveq1d 6908 | . 2 ⊢ (𝜑 → ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)) = (𝑋‘(𝐽 − 1))) |
| 52 | 26 | nn0cni 12538 | . . . . . . 7 ⊢ 𝐿 ∈ ℂ |
| 53 | 52, 43 | pncan3oi 11524 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = 𝐿 |
| 54 | 25 | oveq1i 7441 | . . . . . 6 ⊢ ((𝐿 + 1) − 1) = (𝐽 − 1) |
| 55 | 53, 54 | eqtr3i 2767 | . . . . 5 ⊢ 𝐿 = (𝐽 − 1) |
| 56 | 55 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐿 = (𝐽 − 1)) |
| 57 | 56 | fveq2d 6910 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = (𝑋‘(𝐽 − 1))) |
| 58 | lmatfvlem.8 | . . 3 ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) | |
| 59 | 57, 58 | eqtr3d 2779 | . 2 ⊢ (𝜑 → (𝑋‘(𝐽 − 1)) = 𝑌) |
| 60 | 41, 51, 59 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 litMatclmat 33810 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-lmat 33811 |
| This theorem is referenced by: lmat22e12 33818 lmat22e21 33819 lmat22e22 33820 |
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