Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22e11 | Structured version Visualization version GIF version |
Description: Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-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 |
---|---|
lmat22e11 | ⊢ (𝜑 → (1𝑀1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmat22.m | . . 3 ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) | |
2 | 2nn 11713 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ∈ ℕ) |
4 | lmat22.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | lmat22.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
6 | 4, 5 | s2cld 14235 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
7 | lmat22.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | lmat22.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
9 | 7, 8 | s2cld 14235 | . . . 4 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
10 | 6, 9 | s2cld 14235 | . . 3 ⊢ (𝜑 → 〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉 ∈ Word Word 𝑉) |
11 | s2len 14253 | . . . 4 ⊢ (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2 | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → (♯‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) = 2) |
13 | 1, 4, 5, 7, 8 | lmat22lem 31084 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | 2eluzge1 12297 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘1) | |
15 | eluzfz1 12917 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘1) → 1 ∈ (1...2)) | |
16 | 14, 15 | ax-mp 5 | . . . 4 ⊢ 1 ∈ (1...2) |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (1...2)) |
18 | 1, 3, 10, 12, 13, 17, 17 | lmatfval 31081 | . 2 ⊢ (𝜑 → (1𝑀1) = ((〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘(1 − 1))‘(1 − 1))) |
19 | 1m1e0 11712 | . . . . 5 ⊢ (1 − 1) = 0 | |
20 | 19 | fveq2i 6675 | . . . 4 ⊢ (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘(1 − 1)) = (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) |
21 | s2fv0 14251 | . . . . 5 ⊢ (〈“𝐴𝐵”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
22 | 6, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
23 | 20, 22 | syl5eq 2870 | . . 3 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘(1 − 1)) = 〈“𝐴𝐵”〉) |
24 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → (1 − 1) = 0) |
25 | 23, 24 | fveq12d 6679 | . 2 ⊢ (𝜑 → ((〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘(1 − 1))‘(1 − 1)) = (〈“𝐴𝐵”〉‘0)) |
26 | s2fv0 14251 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵”〉‘0) = 𝐴) | |
27 | 4, 26 | syl 17 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵”〉‘0) = 𝐴) |
28 | 18, 25, 27 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (1𝑀1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 − cmin 10872 ℕcn 11640 2c2 11695 ℤ≥cuz 12246 ...cfz 12895 ♯chash 13693 Word cword 13864 〈“cs2 14205 litMatclmat 31078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-lmat 31079 |
This theorem is referenced by: lmat22det 31089 |
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