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Mirrors > Home > MPE Home > Th. List > wrdl2exs2 | Structured version Visualization version GIF version |
Description: A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.) |
Ref | Expression |
---|---|
wrdl2exs2 | ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠 ∈ 𝑆 ∃𝑡 ∈ 𝑆 𝑊 = 〈“𝑠𝑡”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1le2 11849 | . . . 4 ⊢ 1 ≤ 2 | |
2 | breq2 5072 | . . . 4 ⊢ ((♯‘𝑊) = 2 → (1 ≤ (♯‘𝑊) ↔ 1 ≤ 2)) | |
3 | 1, 2 | mpbiri 260 | . . 3 ⊢ ((♯‘𝑊) = 2 → 1 ≤ (♯‘𝑊)) |
4 | wrdsymb1 13907 | . . 3 ⊢ ((𝑊 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝑊)) → (𝑊‘0) ∈ 𝑆) | |
5 | 3, 4 | sylan2 594 | . 2 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → (𝑊‘0) ∈ 𝑆) |
6 | lsw 13918 | . . . 4 ⊢ (𝑊 ∈ Word 𝑆 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) | |
7 | oveq1 7165 | . . . . . 6 ⊢ ((♯‘𝑊) = 2 → ((♯‘𝑊) − 1) = (2 − 1)) | |
8 | 2m1e1 11766 | . . . . . 6 ⊢ (2 − 1) = 1 | |
9 | 7, 8 | syl6eq 2874 | . . . . 5 ⊢ ((♯‘𝑊) = 2 → ((♯‘𝑊) − 1) = 1) |
10 | 9 | fveq2d 6676 | . . . 4 ⊢ ((♯‘𝑊) = 2 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘1)) |
11 | 6, 10 | sylan9eq 2878 | . . 3 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → (lastS‘𝑊) = (𝑊‘1)) |
12 | 2nn 11713 | . . . 4 ⊢ 2 ∈ ℕ | |
13 | lswlgt0cl 13923 | . . . 4 ⊢ ((2 ∈ ℕ ∧ (𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2)) → (lastS‘𝑊) ∈ 𝑆) | |
14 | 12, 13 | mpan 688 | . . 3 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → (lastS‘𝑊) ∈ 𝑆) |
15 | 11, 14 | eqeltrrd 2916 | . 2 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → (𝑊‘1) ∈ 𝑆) |
16 | wrdlen2s2 14309 | . 2 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → 𝑊 = 〈“(𝑊‘0)(𝑊‘1)”〉) | |
17 | id 22 | . . . . 5 ⊢ (𝑠 = (𝑊‘0) → 𝑠 = (𝑊‘0)) | |
18 | eqidd 2824 | . . . . 5 ⊢ (𝑠 = (𝑊‘0) → 𝑡 = 𝑡) | |
19 | 17, 18 | s2eqd 14227 | . . . 4 ⊢ (𝑠 = (𝑊‘0) → 〈“𝑠𝑡”〉 = 〈“(𝑊‘0)𝑡”〉) |
20 | 19 | eqeq2d 2834 | . . 3 ⊢ (𝑠 = (𝑊‘0) → (𝑊 = 〈“𝑠𝑡”〉 ↔ 𝑊 = 〈“(𝑊‘0)𝑡”〉)) |
21 | eqidd 2824 | . . . . 5 ⊢ (𝑡 = (𝑊‘1) → (𝑊‘0) = (𝑊‘0)) | |
22 | id 22 | . . . . 5 ⊢ (𝑡 = (𝑊‘1) → 𝑡 = (𝑊‘1)) | |
23 | 21, 22 | s2eqd 14227 | . . . 4 ⊢ (𝑡 = (𝑊‘1) → 〈“(𝑊‘0)𝑡”〉 = 〈“(𝑊‘0)(𝑊‘1)”〉) |
24 | 23 | eqeq2d 2834 | . . 3 ⊢ (𝑡 = (𝑊‘1) → (𝑊 = 〈“(𝑊‘0)𝑡”〉 ↔ 𝑊 = 〈“(𝑊‘0)(𝑊‘1)”〉)) |
25 | 20, 24 | rspc2ev 3637 | . 2 ⊢ (((𝑊‘0) ∈ 𝑆 ∧ (𝑊‘1) ∈ 𝑆 ∧ 𝑊 = 〈“(𝑊‘0)(𝑊‘1)”〉) → ∃𝑠 ∈ 𝑆 ∃𝑡 ∈ 𝑆 𝑊 = 〈“𝑠𝑡”〉) |
26 | 5, 15, 16, 25 | syl3anc 1367 | 1 ⊢ ((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠 ∈ 𝑆 ∃𝑡 ∈ 𝑆 𝑊 = 〈“𝑠𝑡”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 ≤ cle 10678 − cmin 10872 ℕcn 11640 2c2 11695 ♯chash 13693 Word cword 13864 lastSclsw 13916 〈“cs2 14205 |
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-lsw 13917 df-concat 13925 df-s1 13952 df-s2 14212 |
This theorem is referenced by: (None) |
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