Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > usgr2wlkspthlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for usgr2wlkspth 27539. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) |
Ref | Expression |
---|---|
usgr2wlkspthlem2 | ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun ◡𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐺 ∈ USGraph) | |
2 | 1 | anim2i 618 | . . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ USGraph)) |
3 | 2 | ancomd 464 | . . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (𝐺 ∈ USGraph ∧ 𝐹(Walks‘𝐺)𝑃)) |
4 | 3simpc 1146 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) | |
5 | 4 | adantl 484 | . . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → ((♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) |
6 | usgr2wlkneq 27536 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1))) | |
7 | 3, 5, 6 | syl2anc 586 | . . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1))) |
8 | simpl 485 | . . . 4 ⊢ ((((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))) | |
9 | fvex 6682 | . . . . 5 ⊢ (𝑃‘0) ∈ V | |
10 | fvex 6682 | . . . . 5 ⊢ (𝑃‘1) ∈ V | |
11 | fvex 6682 | . . . . 5 ⊢ (𝑃‘2) ∈ V | |
12 | 9, 10, 11 | 3pm3.2i 1335 | . . . 4 ⊢ ((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V) |
13 | 8, 12 | jctil 522 | . . 3 ⊢ ((((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1)) → (((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))) |
14 | funcnvs3 14275 | . . 3 ⊢ ((((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))) → Fun ◡〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉) | |
15 | 7, 13, 14 | 3syl 18 | . 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun ◡〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉) |
16 | eqid 2821 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
17 | 16 | wlkpwrd 27398 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺)) |
18 | wlklenvp1 27399 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
19 | oveq1 7162 | . . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → ((♯‘𝐹) + 1) = (2 + 1)) | |
20 | 2p1e3 11778 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
21 | 19, 20 | syl6eq 2872 | . . . . . . 7 ⊢ ((♯‘𝐹) = 2 → ((♯‘𝐹) + 1) = 3) |
22 | 21 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) + 1) = 3) |
23 | 18, 22 | sylan9eq 2876 | . . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (♯‘𝑃) = 3) |
24 | wrdlen3s3 14310 | . . . . 5 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = 3) → 𝑃 = 〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉) | |
25 | 17, 23, 24 | syl2an2r 683 | . . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → 𝑃 = 〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉) |
26 | 25 | cnveqd 5745 | . . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → ◡𝑃 = ◡〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉) |
27 | 26 | funeqd 6376 | . 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (Fun ◡𝑃 ↔ Fun ◡〈“(𝑃‘0)(𝑃‘1)(𝑃‘2)”〉)) |
28 | 15, 27 | mpbird 259 | 1 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun ◡𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 class class class wbr 5065 ◡ccnv 5553 Fun wfun 6348 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 2c2 11691 3c3 11692 ♯chash 13689 Word cword 13860 〈“cs3 14203 Vtxcvtx 26780 USGraphcusgr 26933 Walkscwlks 27377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-edg 26832 df-uhgr 26842 df-upgr 26866 df-umgr 26867 df-uspgr 26934 df-usgr 26935 df-wlks 27380 |
This theorem is referenced by: usgr2wlkspth 27539 |
Copyright terms: Public domain | W3C validator |