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Mirrors > Home > MPE Home > Th. List > wwlknvtx | Structured version Visualization version GIF version |
Description: The symbols of a word ð representing a walk of a fixed length ð are vertices. (Contributed by AV, 16-Mar-2022.) |
Ref | Expression |
---|---|
wwlknvtx | ⢠(ð â (ð WWalksN ðº) â âð â (0...ð)(ðâð) â (Vtxâðº)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknbp1 29533 | . 2 ⢠(ð â (ð WWalksN ðº) â (ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1))) | |
2 | simp2 1134 | . . . 4 ⢠((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â ð â Word (Vtxâðº)) | |
3 | nn0z 12579 | . . . . . . . . 9 ⢠(ð â â0 â ð â â€) | |
4 | fzval3 13697 | . . . . . . . . 9 ⢠(ð â †â (0...ð) = (0..^(ð + 1))) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⢠(ð â â0 â (0...ð) = (0..^(ð + 1))) |
6 | 5 | 3ad2ant1 1130 | . . . . . . 7 ⢠((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â (0...ð) = (0..^(ð + 1))) |
7 | 6 | eleq2d 2811 | . . . . . 6 ⢠((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â (ð â (0...ð) â ð â (0..^(ð + 1)))) |
8 | 7 | biimpa 476 | . . . . 5 ⢠(((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â§ ð â (0...ð)) â ð â (0..^(ð + 1))) |
9 | oveq2 7409 | . . . . . . . 8 ⢠((â¯âð) = (ð + 1) â (0..^(â¯âð)) = (0..^(ð + 1))) | |
10 | 9 | eleq2d 2811 | . . . . . . 7 ⢠((â¯âð) = (ð + 1) â (ð â (0..^(â¯âð)) â ð â (0..^(ð + 1)))) |
11 | 10 | 3ad2ant3 1132 | . . . . . 6 ⢠((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â (ð â (0..^(â¯âð)) â ð â (0..^(ð + 1)))) |
12 | 11 | adantr 480 | . . . . 5 ⢠(((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â§ ð â (0...ð)) â (ð â (0..^(â¯âð)) â ð â (0..^(ð + 1)))) |
13 | 8, 12 | mpbird 257 | . . . 4 ⢠(((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â§ ð â (0...ð)) â ð â (0..^(â¯âð))) |
14 | wrdsymbcl 14473 | . . . 4 ⢠((ð â Word (Vtxâðº) â§ ð â (0..^(â¯âð))) â (ðâð) â (Vtxâðº)) | |
15 | 2, 13, 14 | syl2an2r 682 | . . 3 ⢠(((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â§ ð â (0...ð)) â (ðâð) â (Vtxâðº)) |
16 | 15 | ralrimiva 3138 | . 2 ⢠((ð â â0 â§ ð â Word (Vtxâðº) â§ (â¯âð) = (ð + 1)) â âð â (0...ð)(ðâð) â (Vtxâðº)) |
17 | 1, 16 | syl 17 | 1 ⢠(ð â (ð WWalksN ðº) â âð â (0...ð)(ðâð) â (Vtxâðº)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 â§ wa 395 â§ w3a 1084 = wceq 1533 â wcel 2098 âwral 3053 âcfv 6533 (class class class)co 7401 0cc0 11105 1c1 11106 + caddc 11108 â0cn0 12468 â€cz 12554 ...cfz 13480 ..^cfzo 13623 â¯chash 14286 Word cword 14460 Vtxcvtx 28691 WWalksN cwwlksn 29515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-wwlks 29519 df-wwlksn 29520 |
This theorem is referenced by: wwlknllvtx 29535 |
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