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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 30392 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 478 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0) | |
2 | 1 | fveq2d 6413 | . . . . . 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 13953 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
6 | s2fv0 13969 | . . . . . . . 8 ⊢ (〈“𝐴𝐵”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
8 | 7 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
9 | 2, 8 | eqtrd 2831 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐴𝐵”〉) |
10 | 9 | fveq2d 6413 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐴𝐵”〉)) |
11 | s2len 13971 | . . . 4 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
12 | 10, 11 | syl6eq 2847 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
13 | 12 | adantlr 707 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | simpr 478 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1) | |
15 | 14 | fveq2d 6413 | . . . . . 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 13953 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
19 | s2fv1 13970 | . . . . . . . 8 ⊢ (〈“𝐶𝐷”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
21 | 20 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
22 | 15, 21 | eqtrd 2831 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐶𝐷”〉) |
23 | 22 | fveq2d 6413 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐶𝐷”〉)) |
24 | s2len 13971 | . . . 4 ⊢ (♯‘〈“𝐶𝐷”〉) = 2 | |
25 | 23, 24 | syl6eq 2847 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
26 | 25 | adantlr 707 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
27 | fzo0to2pr 12804 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
28 | 27 | eleq2i 2868 | . . . . 5 ⊢ (𝑖 ∈ (0..^2) ↔ 𝑖 ∈ {0, 1}) |
29 | vex 3386 | . . . . . 6 ⊢ 𝑖 ∈ V | |
30 | 29 | elpr 4389 | . . . . 5 ⊢ (𝑖 ∈ {0, 1} ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
31 | 28, 30 | bitri 267 | . . . 4 ⊢ (𝑖 ∈ (0..^2) ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
32 | 31 | biimpi 208 | . . 3 ⊢ (𝑖 ∈ (0..^2) → (𝑖 = 0 ∨ 𝑖 = 1)) |
33 | 32 | adantl 474 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (𝑖 = 0 ∨ 𝑖 = 1)) |
34 | 13, 26, 33 | mpjaodan 982 | 1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 {cpr 4368 ‘cfv 6099 (class class class)co 6876 0cc0 10222 1c1 10223 2c2 11364 ..^cfzo 12716 ♯chash 13366 Word cword 13530 〈“cs2 13923 litMatclmat 30385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-fzo 12717 df-hash 13367 df-word 13531 df-concat 13587 df-s1 13612 df-s2 13930 |
This theorem is referenced by: lmat22e11 30392 lmat22e12 30393 lmat22e21 30394 lmat22e22 30395 lmat22det 30396 |
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