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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22lem | Structured version Visualization version GIF version | ||
| Description: Lemma for lmat22e11 33817 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 14910 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
| 6 | s2fv0 14926 | . . . . . . . 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 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐴𝐵”〉) |
| 10 | 9 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐴𝐵”〉)) |
| 11 | s2len 14928 | . . . 4 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
| 12 | 10, 11 | eqtrdi 2793 | . . 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 14910 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
| 19 | s2fv1 14927 | . . . . . . . 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 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐶𝐷”〉) |
| 23 | 22 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐶𝐷”〉)) |
| 24 | s2len 14928 | . . . 4 ⊢ (♯‘〈“𝐶𝐷”〉) = 2 | |
| 25 | 23, 24 | eqtrdi 2793 | . . 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 13789 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
| 28 | 27 | eleq2i 2833 | . . . . 5 ⊢ (𝑖 ∈ (0..^2) ↔ 𝑖 ∈ {0, 1}) |
| 29 | vex 3484 | . . . . . 6 ⊢ 𝑖 ∈ V | |
| 30 | 29 | elpr 4650 | . . . . 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 961 | 1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 {cpr 4628 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 2c2 12321 ..^cfzo 13694 ♯chash 14369 Word cword 14552 〈“cs2 14880 litMatclmat 33810 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 |
| This theorem is referenced by: lmat22e11 33817 lmat22e12 33818 lmat22e21 33819 lmat22e22 33820 lmat22det 33821 |
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