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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22lem | Structured version Visualization version GIF version |
Description: Lemma for lmat22e11 33778 and co. (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 |
---|---|
lmat22lem | ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0) | |
2 | 1 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0)) |
3 | lmat22.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | lmat22.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
5 | 3, 4 | s2cld 14906 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
6 | s2fv0 14922 | . . . . . . . 8 ⊢ (〈“𝐴𝐵”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
9 | 2, 8 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐴𝐵”〉) |
10 | 9 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐴𝐵”〉)) |
11 | s2len 14924 | . . . 4 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
12 | 10, 11 | eqtrdi 2790 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
13 | 12 | adantlr 715 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1) | |
15 | 14 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1)) |
16 | lmat22.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
17 | lmat22.d | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
18 | 16, 17 | s2cld 14906 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
19 | s2fv1 14923 | . . . . . . . 8 ⊢ (〈“𝐶𝐷”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
22 | 15, 21 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐶𝐷”〉) |
23 | 22 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐶𝐷”〉)) |
24 | s2len 14924 | . . . 4 ⊢ (♯‘〈“𝐶𝐷”〉) = 2 | |
25 | 23, 24 | eqtrdi 2790 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
26 | 25 | adantlr 715 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
27 | fzo0to2pr 13785 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
28 | 27 | eleq2i 2830 | . . . . 5 ⊢ (𝑖 ∈ (0..^2) ↔ 𝑖 ∈ {0, 1}) |
29 | vex 3481 | . . . . . 6 ⊢ 𝑖 ∈ V | |
30 | 29 | elpr 4654 | . . . . 5 ⊢ (𝑖 ∈ {0, 1} ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
31 | 28, 30 | bitri 275 | . . . 4 ⊢ (𝑖 ∈ (0..^2) ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
32 | 31 | biimpi 216 | . . 3 ⊢ (𝑖 ∈ (0..^2) → (𝑖 = 0 ∨ 𝑖 = 1)) |
33 | 32 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (𝑖 = 0 ∨ 𝑖 = 1)) |
34 | 13, 26, 33 | mpjaodan 960 | 1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 {cpr 4632 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 2c2 12318 ..^cfzo 13690 ♯chash 14365 Word cword 14548 〈“cs2 14876 litMatclmat 33771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 |
This theorem is referenced by: lmat22e11 33778 lmat22e12 33779 lmat22e21 33780 lmat22e22 33781 lmat22det 33782 |
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