![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s2len | Structured version Visualization version GIF version |
Description: The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s2len | ⊢ (♯‘〈“𝐴𝐵”〉) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s2 14046 | . 2 ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++ 〈“𝐵”〉) | |
2 | s1cli 13803 | . 2 ⊢ 〈“𝐴”〉 ∈ Word V | |
3 | s1len 13804 | . 2 ⊢ (♯‘〈“𝐴”〉) = 1 | |
4 | 1p1e2 11610 | . 2 ⊢ (1 + 1) = 2 | |
5 | 1, 2, 3, 4 | cats1len 14058 | 1 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ‘cfv 6225 1c1 10384 2c2 11540 ♯chash 13540 〈“cs1 13793 〈“cs2 14039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-s2 14046 |
This theorem is referenced by: s2dm 14088 s3fv0 14089 s3fv1 14090 s3fv2 14091 s3len 14092 lsws2 14102 s3tpop 14107 s4prop 14108 s3eqs2s1eq 14136 pfx2 14145 psgnunilem2 18354 efgtlen 18579 efgredleme 18596 efgredlemc 18598 frgpnabllem1 18716 2wlkdlem1 27391 2wlkdlem2 27392 2wlkdlem4 27394 2pthdlem1 27396 2wlkond 27403 2pthd 27406 2pthon3v 27409 umgr2adedgwlk 27411 s2elclwwlknon2 27570 1wlkdlem1 27603 wlk2v2e 27623 pfx1s2 30299 s2rn 30300 cshw1s2 30308 cyc2fv1 30410 cyc2fv2 30411 lmat22lem 30697 lmat22e11 30698 lmat22e12 30699 lmat22e21 30700 lmat22e22 30701 lmat22det 30702 fiblem 31273 fib0 31274 fib1 31275 fibp1 31276 2cycld 31993 umgr2cycl 31996 amgm2d 40037 amgmw2d 44385 |
Copyright terms: Public domain | W3C validator |