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Mirrors > Home > MPE Home > Th. List > cats1cat | Structured version Visualization version GIF version |
Description: Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
cats1cld.1 | ⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩) |
cats1cat.2 | ⊢ 𝐴 ∈ Word V |
cats1cat.3 | ⊢ 𝑆 ∈ Word V |
cats1cat.4 | ⊢ 𝐶 = (𝐵 ++ ⟨“𝑋”⟩) |
cats1cat.5 | ⊢ 𝐵 = (𝐴 ++ 𝑆) |
Ref | Expression |
---|---|
cats1cat | ⊢ 𝐶 = (𝐴 ++ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cats1cat.5 | . . . 4 ⊢ 𝐵 = (𝐴 ++ 𝑆) | |
2 | 1 | oveq1i 7347 | . . 3 ⊢ (𝐵 ++ ⟨“𝑋”⟩) = ((𝐴 ++ 𝑆) ++ ⟨“𝑋”⟩) |
3 | cats1cat.2 | . . . 4 ⊢ 𝐴 ∈ Word V | |
4 | cats1cat.3 | . . . 4 ⊢ 𝑆 ∈ Word V | |
5 | s1cli 14409 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
6 | ccatass 14392 | . . . 4 ⊢ ((𝐴 ∈ Word V ∧ 𝑆 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V) → ((𝐴 ++ 𝑆) ++ ⟨“𝑋”⟩) = (𝐴 ++ (𝑆 ++ ⟨“𝑋”⟩))) | |
7 | 3, 4, 5, 6 | mp3an 1460 | . . 3 ⊢ ((𝐴 ++ 𝑆) ++ ⟨“𝑋”⟩) = (𝐴 ++ (𝑆 ++ ⟨“𝑋”⟩)) |
8 | 2, 7 | eqtri 2764 | . 2 ⊢ (𝐵 ++ ⟨“𝑋”⟩) = (𝐴 ++ (𝑆 ++ ⟨“𝑋”⟩)) |
9 | cats1cat.4 | . 2 ⊢ 𝐶 = (𝐵 ++ ⟨“𝑋”⟩) | |
10 | cats1cld.1 | . . 3 ⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩) | |
11 | 10 | oveq2i 7348 | . 2 ⊢ (𝐴 ++ 𝑇) = (𝐴 ++ (𝑆 ++ ⟨“𝑋”⟩)) |
12 | 8, 9, 11 | 3eqtr4i 2774 | 1 ⊢ 𝐶 = (𝐴 ++ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3441 (class class class)co 7337 Word cword 14317 ++ cconcat 14373 ⟨“cs1 14399 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-concat 14374 df-s1 14400 |
This theorem is referenced by: s1s2 14735 s1s3 14736 s1s4 14737 s1s5 14738 s1s6 14739 s1s7 14740 s2s2 14741 s4s2 14742 s4s3 14743 s4s4 14744 |
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