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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e12 | Structured version Visualization version GIF version |
Description: Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
Ref | Expression |
---|---|
lmat22.m | ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) |
lmat22.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
lmat22.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
lmat22.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
lmat22.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
Ref | Expression |
---|---|
lmat22e12 | ⊢ (𝜑 → (1𝑀2) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . 2 ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) | |
2 | 2nn 11791 | . . 3 ⊢ 2 ∈ ℕ | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 2 ∈ ℕ) |
4 | lmat22.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | lmat22.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
6 | 4, 5 | s2cld 14324 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
7 | lmat22.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | lmat22.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
9 | 7, 8 | s2cld 14324 | . . 3 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
10 | 6, 9 | s2cld 14324 | . 2 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
11 | s2len 14342 | . . 3 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 31341 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | 0nn0 11993 | . 2 ⊢ 0 ∈ ℕ0 | |
15 | 1nn0 11994 | . 2 ⊢ 1 ∈ ℕ0 | |
16 | 1le2 11927 | . 2 ⊢ 1 ≤ 2 | |
17 | 2 | nnrei 11727 | . . 3 ⊢ 2 ∈ ℝ |
18 | 17 | leidi 11254 | . 2 ⊢ 2 ≤ 2 |
19 | 0p1e1 11840 | . 2 ⊢ (0 + 1) = 1 | |
20 | 1p1e2 11843 | . 2 ⊢ (1 + 1) = 2 | |
21 | s2cli 14333 | . . 3 ⊢ 〈“𝐴𝐵”〉 ∈ Word V | |
22 | s2fv0 14340 | . . 3 ⊢ (〈“𝐴𝐵”〉 ∈ Word V → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
23 | 21, 22 | ax-mp 5 | . 2 ⊢ (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉 |
24 | s2fv1 14341 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵”〉‘1) = 𝐵) | |
25 | 5, 24 | syl 17 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉‘1) = 𝐵) |
26 | 1, 3, 10, 12, 13, 14, 15, 16, 18, 19, 20, 23, 25 | lmatfvlem 31339 | 1 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ‘cfv 6339 (class class class)co 7172 0cc0 10617 1c1 10618 ℕcn 11718 2c2 11773 ♯chash 13784 Word cword 13957 〈“cs2 14294 litMatclmat 31335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-fzo 13127 df-hash 13785 df-word 13958 df-concat 14014 df-s1 14041 df-s2 14301 df-lmat 31336 |
This theorem is referenced by: lmat22det 31346 |
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