![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14562 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
2 | s1cli 14562 | . . . 4 ⊢ 〈“𝑌”〉 ∈ Word V | |
3 | 1z 12599 | . . . . . 6 ⊢ 1 ∈ ℤ | |
4 | 2z 12601 | . . . . . 6 ⊢ 2 ∈ ℤ | |
5 | 1lt2 12390 | . . . . . 6 ⊢ 1 < 2 | |
6 | fzolb 13645 | . . . . . 6 ⊢ (1 ∈ (1..^2) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 < 2)) | |
7 | 3, 4, 5, 6 | mpbir3an 1340 | . . . . 5 ⊢ 1 ∈ (1..^2) |
8 | s1len 14563 | . . . . . 6 ⊢ (♯‘〈“𝑋”〉) = 1 | |
9 | s1len 14563 | . . . . . . . 8 ⊢ (♯‘〈“𝑌”〉) = 1 | |
10 | 8, 9 | oveq12i 7424 | . . . . . . 7 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = (1 + 1) |
11 | 1p1e2 12344 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
12 | 10, 11 | eqtri 2759 | . . . . . 6 ⊢ ((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)) = 2 |
13 | 8, 12 | oveq12i 7424 | . . . . 5 ⊢ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) = (1..^2) |
14 | 7, 13 | eleqtrri 2831 | . . . 4 ⊢ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉))) |
15 | ccatval2 14535 | . . . 4 ⊢ ((〈“𝑋”〉 ∈ Word V ∧ 〈“𝑌”〉 ∈ Word V ∧ 1 ∈ ((♯‘〈“𝑋”〉)..^((♯‘〈“𝑋”〉) + (♯‘〈“𝑌”〉)))) → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉)))) | |
16 | 1, 2, 14, 15 | mp3an 1460 | . . 3 ⊢ ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) |
17 | 8 | oveq2i 7423 | . . . . 5 ⊢ (1 − (♯‘〈“𝑋”〉)) = (1 − 1) |
18 | 1m1e0 12291 | . . . . 5 ⊢ (1 − 1) = 0 | |
19 | 17, 18 | eqtri 2759 | . . . 4 ⊢ (1 − (♯‘〈“𝑋”〉)) = 0 |
20 | 19 | fveq2i 6894 | . . 3 ⊢ (〈“𝑌”〉‘(1 − (♯‘〈“𝑋”〉))) = (〈“𝑌”〉‘0) |
21 | 16, 20 | eqtri 2759 | . 2 ⊢ ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = (〈“𝑌”〉‘0) |
22 | s1fv 14567 | . 2 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑌”〉‘0) = 𝑌) | |
23 | 21, 22 | eqtrid 2783 | 1 ⊢ (𝑌 ∈ 𝑉 → ((〈“𝑋”〉 ++ 〈“𝑌”〉)‘1) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 < clt 11255 − cmin 11451 2c2 12274 ℤcz 12565 ..^cfzo 13634 ♯chash 14297 Word cword 14471 ++ cconcat 14527 〈“cs1 14552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 |
This theorem is referenced by: tworepnotupword 46062 |
Copyright terms: Public domain | W3C validator |