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Mirrors > Home > MPE Home > Th. List > ccat2s1p2 | Structured version Visualization version GIF version |
Description: Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.) |
Ref | Expression |
---|---|
ccat2s1p2 | ⊢ (𝑌 ∈ 𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cli 14355 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
2 | s1cli 14355 | . . . 4 ⊢ ⟨“𝑌”⟩ ∈ Word V | |
3 | 1z 12396 | . . . . . 6 ⊢ 1 ∈ ℤ | |
4 | 2z 12398 | . . . . . 6 ⊢ 2 ∈ ℤ | |
5 | 1lt2 12190 | . . . . . 6 ⊢ 1 < 2 | |
6 | fzolb 13439 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
7 | 3, 4, 5, 6 | mpbir3an 1341 | . . . . 5 ⊢ 1 ∈ (1..^2) |
8 | s1len 14356 | . . . . . 6 ⊢ (♯‘⟨“𝑋”⟩) = 1 | |
9 | s1len 14356 | . . . . . . . 8 ⊢ (♯‘⟨“𝑌”⟩) = 1 | |
10 | 8, 9 | oveq12i 7319 | . . . . . . 7 ⊢ ((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)) = (1 + 1) |
11 | 1p1e2 12144 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtri 2764 | . . . . . 6 ⊢ ((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)) = 2 |
13 | 8, 12 | oveq12i 7319 | . . . . 5 ⊢ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩))) = (1..^2) |
14 | 7, 13 | eleqtrri 2836 | . . . 4 ⊢ 1 ∈ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩))) |
15 | ccatval2 14328 | . . . 4 ⊢ ((⟨“𝑋”⟩ ∈ Word V ∧ ⟨“𝑌”⟩ ∈ Word V ∧ 1 ∈ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)))) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩)))) | |
16 | 1, 2, 14, 15 | mp3an 1461 | . . 3 ⊢ ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩))) |
17 | 8 | oveq2i 7318 | . . . . 5 ⊢ (1 − (♯‘⟨“𝑋”⟩)) = (1 − 1) |
18 | 1m1e0 12091 | . . . . 5 ⊢ (1 − 1) = 0 | |
19 | 17, 18 | eqtri 2764 | . . . 4 ⊢ (1 − (♯‘⟨“𝑋”⟩)) = 0 |
20 | 19 | fveq2i 6807 | . . 3 ⊢ (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩))) = (⟨“𝑌”⟩‘0) |
21 | 16, 20 | eqtri 2764 | . 2 ⊢ ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘0) |
22 | s1fv 14360 | . 2 ⊢ (𝑌 ∈ 𝑉 → (⟨“𝑌”⟩‘0) = 𝑌) | |
23 | 21, 22 | eqtrid 2788 | 1 ⊢ (𝑌 ∈ 𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 Vcvv 3437 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 0cc0 10917 1c1 10918 + caddc 10920 < clt 11055 − cmin 11251 2c2 12074 ℤcz 12365 ..^cfzo 13428 ♯chash 14090 Word cword 14262 ++ cconcat 14318 ⟨“cs1 14345 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-hash 14091 df-word 14263 df-concat 14319 df-s1 14346 |
This theorem is referenced by: tworepnotupword 46579 |
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