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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 29212 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 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0) | |
2 | 1 | fveq2d 6107 | . . . . . 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 13466 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
6 | s2fv0 13482 | . . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐴𝐵”〉) |
10 | 9 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 0) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (#‘〈“𝐴𝐵”〉)) |
11 | s2len 13484 | . . . 4 ⊢ (#‘〈“𝐴𝐵”〉) = 2 | |
12 | 10, 11 | syl6eq 2660 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 0) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
13 | 12 | adantlr 747 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 0) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | simpr 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1) | |
15 | 14 | fveq2d 6107 | . . . . . 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 13466 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
19 | s2fv1 13483 | . . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐶𝐷”〉) |
23 | 22 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 1) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (#‘〈“𝐶𝐷”〉)) |
24 | s2len 13484 | . . . 4 ⊢ (#‘〈“𝐶𝐷”〉) = 2 | |
25 | 23, 24 | syl6eq 2660 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 1) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
26 | 25 | adantlr 747 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 1) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
27 | fzo0to2pr 12420 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
28 | 27 | eleq2i 2680 | . . . . 5 ⊢ (𝑖 ∈ (0..^2) ↔ 𝑖 ∈ {0, 1}) |
29 | vex 3176 | . . . . . 6 ⊢ 𝑖 ∈ V | |
30 | 29 | elpr 4146 | . . . . 5 ⊢ (𝑖 ∈ {0, 1} ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
31 | 28, 30 | bitri 263 | . . . 4 ⊢ (𝑖 ∈ (0..^2) ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
32 | 31 | biimpi 205 | . . 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 823 | 1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (#‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cpr 4127 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 2c2 10947 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs2 13437 litMatclmat 29205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 |
This theorem is referenced by: lmat22e11 29212 lmat22e12 29213 lmat22e21 29214 lmat22e22 29215 lmat22det 29216 |
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