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Mirrors > Home > MPE Home > Th. List > umgr2cwwkdifex | Structured version Visualization version GIF version |
Description: If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
umgr2cwwkdifex | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) ≠ (𝑊‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2b2 12854 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
2 | 1nn0 12437 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → 1 ∈ ℕ0) |
4 | simpl 484 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → 𝑁 ∈ ℕ) | |
5 | simpr 486 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → 1 < 𝑁) | |
6 | elfzo0 13622 | . . . . 5 ⊢ (1 ∈ (0..^𝑁) ↔ (1 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
7 | 3, 4, 5, 6 | syl3anbrc 1344 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → 1 ∈ (0..^𝑁)) |
8 | 1, 7 | sylbi 216 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 ∈ (0..^𝑁)) |
9 | 8 | 3ad2ant2 1135 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 1 ∈ (0..^𝑁)) |
10 | fveq2 6846 | . . . 4 ⊢ (𝑖 = 1 → (𝑊‘𝑖) = (𝑊‘1)) | |
11 | 10 | adantl 483 | . . 3 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ 𝑖 = 1) → (𝑊‘𝑖) = (𝑊‘1)) |
12 | 11 | neeq1d 3000 | . 2 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ 𝑖 = 1) → ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ (𝑊‘1) ≠ (𝑊‘0))) |
13 | umgr2cwwk2dif 29057 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘1) ≠ (𝑊‘0)) | |
14 | 9, 12, 13 | rspcedvd 3585 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) ≠ (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 < clt 11197 ℕcn 12161 2c2 12216 ℕ0cn0 12421 ℤ≥cuz 12771 ..^cfzo 13576 UMGraphcumgr 28081 ClWWalksN cclwwlkn 29017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-edg 28048 df-umgr 28083 df-clwwlk 28975 df-clwwlkn 29018 |
This theorem is referenced by: umgrhashecclwwlk 29071 |
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