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Mirrors > Home > MPE Home > Th. List > repswsymballbi | Structured version Visualization version Unicode version |
Description: A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.) |
Ref | Expression |
---|---|
repswsymballbi | Word repeatS ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . 5 | |
2 | hash0 13158 | . . . . 5 | |
3 | 1, 2 | syl6eq 2672 | . . . 4 |
4 | fvex 6201 | . . . . . . . 8 | |
5 | repsw0 13524 | . . . . . . . 8 repeatS | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 repeatS |
7 | 6 | eqcomi 2631 | . . . . . 6 repeatS |
8 | simpr 477 | . . . . . 6 | |
9 | oveq2 6658 | . . . . . . 7 repeatS repeatS | |
10 | 9 | adantr 481 | . . . . . 6 repeatS repeatS |
11 | 7, 8, 10 | 3eqtr4a 2682 | . . . . 5 repeatS |
12 | ral0 4076 | . . . . . . 7 | |
13 | oveq2 6658 | . . . . . . . . 9 ..^ ..^ | |
14 | fzo0 12492 | . . . . . . . . 9 ..^ | |
15 | 13, 14 | syl6eq 2672 | . . . . . . . 8 ..^ |
16 | 15 | raleqdv 3144 | . . . . . . 7 ..^ |
17 | 12, 16 | mpbiri 248 | . . . . . 6 ..^ |
18 | 17 | adantr 481 | . . . . 5 ..^ |
19 | 11, 18 | 2thd 255 | . . . 4 repeatS ..^ |
20 | 3, 19 | mpancom 703 | . . 3 repeatS ..^ |
21 | 20 | a1d 25 | . 2 Word repeatS ..^ |
22 | df-3an 1039 | . . . . 5 Word ..^ Word ..^ | |
23 | 22 | a1i 11 | . . . 4 Word Word ..^ Word ..^ |
24 | fstwrdne 13344 | . . . . . 6 Word | |
25 | 24 | ancoms 469 | . . . . 5 Word |
26 | lencl 13324 | . . . . . 6 Word | |
27 | 26 | adantl 482 | . . . . 5 Word |
28 | repsdf2 13525 | . . . . 5 repeatS Word ..^ | |
29 | 25, 27, 28 | syl2anc 693 | . . . 4 Word repeatS Word ..^ |
30 | simpr 477 | . . . . . 6 Word Word | |
31 | eqidd 2623 | . . . . . 6 Word | |
32 | 30, 31 | jca 554 | . . . . 5 Word Word |
33 | 32 | biantrurd 529 | . . . 4 Word ..^ Word ..^ |
34 | 23, 29, 33 | 3bitr4d 300 | . . 3 Word repeatS ..^ |
35 | 34 | ex 450 | . 2 Word repeatS ..^ |
36 | 21, 35 | pm2.61ine 2877 | 1 Word repeatS ..^ |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 cvv 3200 c0 3915 cfv 5888 (class class class)co 6650 cc0 9936 cn0 11292 ..^cfzo 12465 chash 13117 Word cword 13291 repeatS creps 13298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-reps 13306 |
This theorem is referenced by: cshw1repsw 13569 cshwsidrepsw 15800 cshwshashlem1 15802 cshwshash 15811 |
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