Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lmat22lem | Structured version Visualization version GIF version |
Description: Lemma for lmat22e11 31085 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 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0) | |
2 | 1 | fveq2d 6676 | . . . . . 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 14235 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑉) |
6 | s2fv0 14251 | . . . . . . . 8 ⊢ (〈“𝐴𝐵”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
8 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘0) = 〈“𝐴𝐵”〉) |
9 | 2, 8 | eqtrd 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 0) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐴𝐵”〉) |
10 | 9 | fveq2d 6676 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐴𝐵”〉)) |
11 | s2len 14253 | . . . 4 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
12 | 10, 11 | syl6eq 2874 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
13 | 12 | adantlr 713 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 0) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
14 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1) | |
15 | 14 | fveq2d 6676 | . . . . . 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 14235 | . . . . . . . 8 ⊢ (𝜑 → 〈“𝐶𝐷”〉 ∈ Word 𝑉) |
19 | s2fv1 14252 | . . . . . . . 8 ⊢ (〈“𝐶𝐷”〉 ∈ Word 𝑉 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
21 | 20 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘1) = 〈“𝐶𝐷”〉) |
22 | 15, 21 | eqtrd 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 1) → (〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖) = 〈“𝐶𝐷”〉) |
23 | 22 | fveq2d 6676 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = (♯‘〈“𝐶𝐷”〉)) |
24 | s2len 14253 | . . . 4 ⊢ (♯‘〈“𝐶𝐷”〉) = 2 | |
25 | 23, 24 | syl6eq 2874 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
26 | 25 | adantlr 713 | . 2 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^2)) ∧ 𝑖 = 1) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
27 | fzo0to2pr 13125 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
28 | 27 | eleq2i 2906 | . . . . 5 ⊢ (𝑖 ∈ (0..^2) ↔ 𝑖 ∈ {0, 1}) |
29 | vex 3499 | . . . . . 6 ⊢ 𝑖 ∈ V | |
30 | 29 | elpr 4592 | . . . . 5 ⊢ (𝑖 ∈ {0, 1} ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
31 | 28, 30 | bitri 277 | . . . 4 ⊢ (𝑖 ∈ (0..^2) ↔ (𝑖 = 0 ∨ 𝑖 = 1)) |
32 | 31 | biimpi 218 | . . 3 ⊢ (𝑖 ∈ (0..^2) → (𝑖 = 0 ∨ 𝑖 = 1)) |
33 | 32 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (𝑖 = 0 ∨ 𝑖 = 1)) |
34 | 13, 26, 33 | mpjaodan 955 | 1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cpr 4571 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 2c2 11695 ..^cfzo 13036 ♯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 |
This theorem is referenced by: lmat22e11 31085 lmat22e12 31086 lmat22e21 31087 lmat22e22 31088 lmat22det 31089 |
Copyright terms: Public domain | W3C validator |