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Mirrors > Home > MPE Home > Th. List > ccat2s1p2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ccat2s1p2 13979 as of 20-Jan-2024. Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ccat2s1p2OLD | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 13949 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈“𝑋”〉 ∈ Word 𝑉) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈“𝑋”〉 ∈ Word 𝑉) |
3 | s1cl 13949 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → 〈“𝑌”〉 ∈ Word 𝑉) | |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈“𝑌”〉 ∈ Word 𝑉) |
5 | 1z 12006 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 2z 12008 | . . . . . 6 ⊢ 2 ∈ ℤ | |
7 | 1lt2 11802 | . . . . . 6 ⊢ 1 < 2 | |
8 | fzolb 13041 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
9 | 5, 6, 7, 8 | mpbir3an 1336 | . . . . 5 ⊢ 1 ∈ (1..^2) |
10 | s1len 13953 | . . . . . 6 ⊢ (♯‘〈“𝑋”〉) = 1 | |
11 | s1len 13953 | . . . . . . . 8 ⊢ (♯‘〈“𝑌”〉) = 1 | |
12 | 10, 11 | oveq12i 7161 | . . . . . . 7 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = (1 + 1) |
13 | 1p1e2 11756 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
14 | 12, 13 | eqtri 2843 | . . . . . 6 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = 2 |
15 | 10, 14 | oveq12i 7161 | . . . . 5 ⊢ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) = (1..^2) |
16 | 9, 15 | eleqtrri 2911 | . . . 4 ⊢ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) |
17 | 16 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)))) |
18 | ccatval2 13925 | . . 3 ⊢ ((〈“𝑋”〉 ∈ Word 𝑉 ∧ 〈“𝑌”〉 ∈ Word 𝑉 ∧ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)))) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉)))) | |
19 | 2, 4, 17, 18 | syl3anc 1366 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉)))) |
20 | 10 | oveq2i 7160 | . . . . . . 7 ⊢ (1 − (♯‘〈“𝑋”〉)) = (1 − 1) |
21 | 1m1e0 11703 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
22 | 20, 21 | eqtri 2843 | . . . . . 6 ⊢ (1 − (♯‘〈“𝑋”〉)) = 0 |
23 | 22 | a1i 11 | . . . . 5 ⊢ (𝑌 ∈ 𝑉 → (1 − (♯‘〈“𝑋”〉)) = 0) |
24 | 23 | fveq2d 6667 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) = (〈“𝑌”〉‘0)) |
25 | s1fv 13957 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘0) = 𝑌) | |
26 | 24, 25 | eqtrd 2855 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) = 𝑌) |
27 | 26 | adantl 484 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) = 𝑌) |
28 | 19, 27 | eqtrd 2855 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 0cc0 10530 1c1 10531 + caddc 10533 < clt 10668 − cmin 10863 2c2 11686 ℤcz 11975 ..^cfzo 13030 ♯chash 13687 Word cword 13858 ++ cconcat 13915 〈“cs1 13942 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13916 df-s1 13943 |
This theorem is referenced by: (None) |
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