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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 14587 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
2 | s1cli 14587 | . . . 4 ⊢ ⟨“𝑌”⟩ ∈ Word V | |
3 | 1z 12622 | . . . . . 6 ⊢ 1 ∈ ℤ | |
4 | 2z 12624 | . . . . . 6 ⊢ 2 ∈ ℤ | |
5 | 1lt2 12413 | . . . . . 6 ⊢ 1 < 2 | |
6 | fzolb 13670 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
7 | 3, 4, 5, 6 | mpbir3an 1338 | . . . . 5 ⊢ 1 ∈ (1..^2) |
8 | s1len 14588 | . . . . . 6 ⊢ (♯‘⟨“𝑋”⟩) = 1 | |
9 | s1len 14588 | . . . . . . . 8 ⊢ (♯‘⟨“𝑌”⟩) = 1 | |
10 | 8, 9 | oveq12i 7428 | . . . . . . 7 ⊢ ((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)) = (1 + 1) |
11 | 1p1e2 12367 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtri 2753 | . . . . . 6 ⊢ ((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)) = 2 |
13 | 8, 12 | oveq12i 7428 | . . . . 5 ⊢ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩))) = (1..^2) |
14 | 7, 13 | eleqtrri 2824 | . . . 4 ⊢ 1 ∈ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩))) |
15 | ccatval2 14560 | . . . 4 ⊢ ((⟨“𝑋”⟩ ∈ Word V ∧ ⟨“𝑌”⟩ ∈ Word V ∧ 1 ∈ ((♯‘⟨“𝑋”⟩)..^((♯‘⟨“𝑋”⟩) + (♯‘⟨“𝑌”⟩)))) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩)))) | |
16 | 1, 2, 14, 15 | mp3an 1457 | . . 3 ⊢ ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩))) |
17 | 8 | oveq2i 7427 | . . . . 5 ⊢ (1 − (♯‘⟨“𝑋”⟩)) = (1 − 1) |
18 | 1m1e0 12314 | . . . . 5 ⊢ (1 − 1) = 0 | |
19 | 17, 18 | eqtri 2753 | . . . 4 ⊢ (1 − (♯‘⟨“𝑋”⟩)) = 0 |
20 | 19 | fveq2i 6895 | . . 3 ⊢ (⟨“𝑌”⟩‘(1 − (♯‘⟨“𝑋”⟩))) = (⟨“𝑌”⟩‘0) |
21 | 16, 20 | eqtri 2753 | . 2 ⊢ ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = (⟨“𝑌”⟩‘0) |
22 | s1fv 14592 | . 2 ⊢ (𝑌 ∈ 𝑉 → (⟨“𝑌”⟩‘0) = 𝑌) | |
23 | 21, 22 | eqtrid 2777 | 1 ⊢ (𝑌 ∈ 𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 < clt 11278 − cmin 11474 2c2 12297 ℤcz 12588 ..^cfzo 13659 ♯chash 14321 Word cword 14496 ++ cconcat 14552 ⟨“cs1 14577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 |
This theorem is referenced by: tworepnotupword 46335 |
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