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Mirrors > Home > MPE Home > Th. List > s3cld | Structured version Visualization version GIF version |
Description: A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
s2cld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
s2cld.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
s3cld.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
s3cld | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s3 13803 | . 2 ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) | |
2 | s2cld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
3 | s2cld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
4 | 2, 3 | s2cld 13825 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑋) |
5 | s3cld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | 1, 4, 5 | cats1cld 13809 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 Word cword 13487 〈“cs2 13795 〈“cs3 13796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-fzo 12674 df-hash 13322 df-word 13495 df-concat 13497 df-s1 13498 df-s2 13802 df-s3 13803 |
This theorem is referenced by: s4cld 13827 s3cl 13833 s4prop 13864 trgcgrg 25631 tgcgr4 25647 israg 25813 iscgra 25922 isinag 25950 isleag 25954 iseqlg 25968 circlemethhgt 31061 gsumws4 39026 amgm3d 39028 |
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