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Mirrors > Home > MPE Home > Th. List > cats2cat | Structured version Visualization version GIF version |
Description: Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) |
Ref | Expression |
---|---|
cats2cat.b | ⊢ 𝐵 ∈ Word V |
cats2cat.d | ⊢ 𝐷 ∈ Word V |
cats2cat.a | ⊢ 𝐴 = (𝐵 ++ 〈“𝑋”〉) |
cats2cat.c | ⊢ 𝐶 = (〈“𝑌”〉 ++ 𝐷) |
Ref | Expression |
---|---|
cats2cat | ⊢ (𝐴 ++ 𝐶) = ((𝐵 ++ 〈“𝑋𝑌”〉) ++ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cats2cat.a | . . 3 ⊢ 𝐴 = (𝐵 ++ 〈“𝑋”〉) | |
2 | cats2cat.c | . . 3 ⊢ 𝐶 = (〈“𝑌”〉 ++ 𝐷) | |
3 | 1, 2 | oveq12i 7442 | . 2 ⊢ (𝐴 ++ 𝐶) = ((𝐵 ++ 〈“𝑋”〉) ++ (〈“𝑌”〉 ++ 𝐷)) |
4 | cats2cat.b | . . . 4 ⊢ 𝐵 ∈ Word V | |
5 | s1cli 14639 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
6 | ccatcl 14608 | . . . 4 ⊢ ((𝐵 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V) → (𝐵 ++ 〈“𝑋”〉) ∈ Word V) | |
7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ (𝐵 ++ 〈“𝑋”〉) ∈ Word V |
8 | s1cli 14639 | . . 3 ⊢ 〈“𝑌”〉 ∈ Word V | |
9 | cats2cat.d | . . 3 ⊢ 𝐷 ∈ Word V | |
10 | ccatass 14622 | . . 3 ⊢ (((𝐵 ++ 〈“𝑋”〉) ∈ Word V ∧ 〈“𝑌”〉 ∈ Word V ∧ 𝐷 ∈ Word V) → (((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ++ 𝐷) = ((𝐵 ++ 〈“𝑋”〉) ++ (〈“𝑌”〉 ++ 𝐷))) | |
11 | 7, 8, 9, 10 | mp3an 1460 | . 2 ⊢ (((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ++ 𝐷) = ((𝐵 ++ 〈“𝑋”〉) ++ (〈“𝑌”〉 ++ 𝐷)) |
12 | ccatass 14622 | . . . . 5 ⊢ ((𝐵 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V ∧ 〈“𝑌”〉 ∈ Word V) → ((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝐵 ++ (〈“𝑋”〉 ++ 〈“𝑌”〉))) | |
13 | 4, 5, 8, 12 | mp3an 1460 | . . . 4 ⊢ ((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝐵 ++ (〈“𝑋”〉 ++ 〈“𝑌”〉)) |
14 | df-s2 14883 | . . . . . 6 ⊢ 〈“𝑋𝑌”〉 = (〈“𝑋”〉 ++ 〈“𝑌”〉) | |
15 | 14 | eqcomi 2743 | . . . . 5 ⊢ (〈“𝑋”〉 ++ 〈“𝑌”〉) = 〈“𝑋𝑌”〉 |
16 | 15 | oveq2i 7441 | . . . 4 ⊢ (𝐵 ++ (〈“𝑋”〉 ++ 〈“𝑌”〉)) = (𝐵 ++ 〈“𝑋𝑌”〉) |
17 | 13, 16 | eqtri 2762 | . . 3 ⊢ ((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝐵 ++ 〈“𝑋𝑌”〉) |
18 | 17 | oveq1i 7440 | . 2 ⊢ (((𝐵 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ++ 𝐷) = ((𝐵 ++ 〈“𝑋𝑌”〉) ++ 𝐷) |
19 | 3, 11, 18 | 3eqtr2i 2768 | 1 ⊢ (𝐴 ++ 𝐶) = ((𝐵 ++ 〈“𝑋𝑌”〉) ++ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 Vcvv 3477 (class class class)co 7430 Word cword 14548 ++ cconcat 14604 〈“cs1 14629 〈“cs2 14876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 |
This theorem is referenced by: s3s4 14968 s2s5 14969 s5s2 14970 |
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