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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e21 | 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 |
---|---|
lmat22e21 | ⊢ (𝜑 → (2𝑀1) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . 2 ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) | |
2 | 2nn 11709 | . . 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 14232 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
7 | lmat22.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | lmat22.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
9 | 7, 8 | s2cld 14232 | . . 3 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
10 | 6, 9 | s2cld 14232 | . 2 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
11 | s2len 14250 | . . 3 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 31082 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | 1nn0 11912 | . 2 ⊢ 1 ∈ ℕ0 | |
15 | 0nn0 11911 | . 2 ⊢ 0 ∈ ℕ0 | |
16 | 2 | nnrei 11646 | . . 3 ⊢ 2 ∈ ℝ |
17 | 16 | leidi 11173 | . 2 ⊢ 2 ≤ 2 |
18 | 1le2 11845 | . 2 ⊢ 1 ≤ 2 | |
19 | 1p1e2 11761 | . 2 ⊢ (1 + 1) = 2 | |
20 | 0p1e1 11758 | . 2 ⊢ (0 + 1) = 1 | |
21 | s2cli 14241 | . . 3 ⊢ 〈“𝐶𝐷”〉 ∈ Word V | |
22 | s2fv1 14249 | . . 3 ⊢ (〈“𝐶𝐷”〉 ∈ Word V → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) | |
23 | 21, 22 | ax-mp 5 | . 2 ⊢ (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉 |
24 | s2fv0 14248 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐶𝐷”〉‘0) = 𝐶) | |
25 | 7, 24 | syl 17 | . 2 ⊢ (𝜑 → (〈“𝐶𝐷”〉‘0) = 𝐶) |
26 | 1, 3, 10, 12, 13, 14, 15, 17, 18, 19, 20, 23, 25 | lmatfvlem 31080 | 1 ⊢ (𝜑 → (2𝑀1) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 ℕcn 11637 2c2 11691 ♯chash 13689 Word cword 13860 〈“cs2 14202 litMatclmat 31076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-lmat 31077 |
This theorem is referenced by: lmat22det 31087 |
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