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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 11698 | . . 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 14224 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
7 | lmat22.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | lmat22.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
9 | 7, 8 | s2cld 14224 | . . 3 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
10 | 6, 9 | s2cld 14224 | . 2 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
11 | s2len 14242 | . . 3 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 31170 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | 0nn0 11900 | . 2 ⊢ 0 ∈ ℕ0 | |
15 | 1nn0 11901 | . 2 ⊢ 1 ∈ ℕ0 | |
16 | 1le2 11834 | . 2 ⊢ 1 ≤ 2 | |
17 | 2 | nnrei 11634 | . . 3 ⊢ 2 ∈ ℝ |
18 | 17 | leidi 11163 | . 2 ⊢ 2 ≤ 2 |
19 | 0p1e1 11747 | . 2 ⊢ (0 + 1) = 1 | |
20 | 1p1e2 11750 | . 2 ⊢ (1 + 1) = 2 | |
21 | s2cli 14233 | . . 3 ⊢ 〈“𝐴𝐵”〉 ∈ Word V | |
22 | s2fv0 14240 | . . 3 ⊢ (〈“𝐴𝐵”〉 ∈ Word V → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
23 | 21, 22 | ax-mp 5 | . 2 ⊢ (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉 |
24 | s2fv1 14241 | . . 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 31168 | 1 ⊢ (𝜑 → (1𝑀2) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ℕcn 11625 2c2 11680 ♯chash 13686 Word cword 13857 〈“cs2 14194 litMatclmat 31164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-lmat 31165 |
This theorem is referenced by: lmat22det 31175 |
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