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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.) |
Ref | Expression |
---|---|
ccat2s1p2 | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 13235 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈“𝑋”〉 ∈ Word 𝑉) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈“𝑋”〉 ∈ Word 𝑉) |
3 | s1cl 13235 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → 〈“𝑌”〉 ∈ Word 𝑉) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈“𝑌”〉 ∈ Word 𝑉) |
5 | 1z 11284 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 2z 11286 | . . . . . 6 ⊢ 2 ∈ ℤ | |
7 | 1lt2 11071 | . . . . . 6 ⊢ 1 < 2 | |
8 | fzolb 12345 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
9 | 5, 6, 7, 8 | mpbir3an 1237 | . . . . 5 ⊢ 1 ∈ (1..^2) |
10 | s1len 13238 | . . . . . 6 ⊢ (#‘〈“𝑋”〉) = 1 | |
11 | s1len 13238 | . . . . . . . 8 ⊢ (#‘〈“𝑌”〉) = 1 | |
12 | 10, 11 | oveq12i 6561 | . . . . . . 7 ⊢ ((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉)) = (1 + 1) |
13 | 1p1e2 11011 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
14 | 12, 13 | eqtri 2632 | . . . . . 6 ⊢ ((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉)) = 2 |
15 | 10, 14 | oveq12i 6561 | . . . . 5 ⊢ ((#‘〈“𝑋”〉)..^((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉))) = (1..^2) |
16 | 9, 15 | eleqtrri 2687 | . . . 4 ⊢ 1 ∈ ((#‘〈“𝑋”〉)..^((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉))) |
17 | 16 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 1 ∈ ((#‘〈“𝑋”〉)..^((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉)))) |
18 | ccatval2 13215 | . . 3 ⊢ ((〈“𝑋”〉 ∈ Word 𝑉 ∧ 〈“𝑌”〉 ∈ Word 𝑉 ∧ 1 ∈ ((#‘〈“𝑋”〉)..^((#‘〈“𝑋”〉) + (#‘〈“𝑌”〉)))) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (#‘〈“𝑋”〉)))) | |
19 | 2, 4, 17, 18 | syl3anc 1318 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (#‘〈“𝑋”〉)))) |
20 | 10 | oveq2i 6560 | . . . . . . 7 ⊢ (1 − (#‘〈“𝑋”〉)) = (1 − 1) |
21 | 1m1e0 10966 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
22 | 20, 21 | eqtri 2632 | . . . . . 6 ⊢ (1 − (#‘〈“𝑋”〉)) = 0 |
23 | 22 | a1i 11 | . . . . 5 ⊢ (𝑌 ∈ 𝑉 → (1 − (#‘〈“𝑋”〉)) = 0) |
24 | 23 | fveq2d 6107 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘(1 − (#‘〈“𝑋”〉))) = (〈“𝑌”〉‘0)) |
25 | s1fv 13243 | . . . 4 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘0) = 𝑌) | |
26 | 24, 25 | eqtrd 2644 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘(1 − (#‘〈“𝑋”〉))) = 𝑌) |
27 | 26 | adantl 481 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (〈“𝑌”〉‘(1 − (#‘〈“𝑋”〉))) = 𝑌) |
28 | 19, 27 | eqtrd 2644 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 2c2 10947 ℤcz 11254 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 |
This theorem is referenced by: (None) |
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