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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 14643 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
| 2 | s1cli 14643 | . . . 4 ⊢ 〈“𝑌”〉 ∈ Word V | |
| 3 | 1z 12647 | . . . . . 6 ⊢ 1 ∈ ℤ | |
| 4 | 2z 12649 | . . . . . 6 ⊢ 2 ∈ ℤ | |
| 5 | 1lt2 12437 | . . . . . 6 ⊢ 1 < 2 | |
| 6 | fzolb 13705 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
| 7 | 3, 4, 5, 6 | mpbir3an 1342 | . . . . 5 ⊢ 1 ∈ (1..^2) |
| 8 | s1len 14644 | . . . . . 6 ⊢ (♯‘〈“𝑋”〉) = 1 | |
| 9 | s1len 14644 | . . . . . . . 8 ⊢ (♯‘〈“𝑌”〉) = 1 | |
| 10 | 8, 9 | oveq12i 7443 | . . . . . . 7 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = (1 + 1) |
| 11 | 1p1e2 12391 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
| 12 | 10, 11 | eqtri 2765 | . . . . . 6 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = 2 |
| 13 | 8, 12 | oveq12i 7443 | . . . . 5 ⊢ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) = (1..^2) |
| 14 | 7, 13 | eleqtrri 2840 | . . . 4 ⊢ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) |
| 15 | ccatval2 14616 | . . . 4 ⊢ ((〈“𝑋”〉 ∈ Word V ∧ 〈“𝑌”〉 ∈ Word V ∧ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)))) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉)))) | |
| 16 | 1, 2, 14, 15 | mp3an 1463 | . . 3 ⊢ ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) |
| 17 | 8 | oveq2i 7442 | . . . . 5 ⊢ (1 − (♯‘〈“𝑋”〉)) = (1 − 1) |
| 18 | 1m1e0 12338 | . . . . 5 ⊢ (1 − 1) = 0 | |
| 19 | 17, 18 | eqtri 2765 | . . . 4 ⊢ (1 − (♯‘〈“𝑋”〉)) = 0 |
| 20 | 19 | fveq2i 6909 | . . 3 ⊢ (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) = (〈“𝑌”〉‘0) |
| 21 | 16, 20 | eqtri 2765 | . 2 ⊢ ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘0) |
| 22 | s1fv 14648 | . 2 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘0) = 𝑌) | |
| 23 | 21, 22 | eqtrid 2789 | 1 ⊢ (𝑌 ∈ 𝑉 → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 − cmin 11492 2c2 12321 ℤcz 12613 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 〈“cs1 14633 |
| 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 |
| This theorem is referenced by: tworepnotupword 46901 |
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