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| Mirrors > Home > MPE Home > Th. List > s2cld | Structured version Visualization version GIF version | ||
| Description: A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
| Ref | Expression |
|---|---|
| s2cld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
| s2cld.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
| Ref | Expression |
|---|---|
| s2cld | ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-s2 14887 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
| 2 | s2cld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 3 | 2 | s1cld 14641 | . 2 ⊢ (𝜑 → 〈“𝐴”〉 ∈ Word 𝑋) |
| 4 | s2cld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
| 5 | 1, 3, 4 | cats1cld 14894 | 1 ⊢ (𝜑 → 〈“𝐴𝐵”〉 ∈ Word 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Word cword 14552 〈“cs1 14633 〈“cs2 14880 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 |
| This theorem is referenced by: s3cld 14911 s2cl 14917 s3co 14960 psgnunilem2 19513 efglem 19734 efgtf 19740 efgtlen 19744 efginvrel2 19745 efgredleme 19761 efgredlemc 19763 efgcpbllemb 19773 frgpuplem 19790 frgpnabllem1 19891 1wlkdlem1 30156 wlk2v2elem1 30174 s2rnOLD 32928 cycpm2tr 33139 cycpm2cl 33140 cyc2fv1 33141 cyc2fv2 33142 cycpmco2 33153 cyc2fvx 33154 cyc3co2 33160 cyc3genpmlem 33171 cyc3genpm 33172 cyc3conja 33177 elrgspnsubrunlem1 33251 lmat22lem 33816 lmat22e11 33817 lmat22e12 33818 lmat22e21 33819 lmat22e22 33820 lmat22det 33821 fib0 34401 fib1 34402 fibp1 34403 gsumws3 44209 amgm2d 44211 amgmw2d 49323 |
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