Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > trlreslem | Structured version Visualization version GIF version |
Description: Lemma for trlres 27485. Formerly part of proof of eupthres 27997. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
trlres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlres.d | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
trlres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
trlres.h | ⊢ 𝐻 = (𝐹 prefix 𝑁) |
Ref | Expression |
---|---|
trlreslem | ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlres.d | . . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
2 | trlres.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | trlf1 27483 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
5 | trlres.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
6 | elfzouz2 13055 | . . . 4 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
7 | fzoss2 13068 | . . . 4 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
9 | f1ores 6632 | . . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) | |
10 | 4, 8, 9 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) |
11 | trlres.h | . . . 4 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
12 | trliswlk 27482 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
13 | 2 | wlkf 27399 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
14 | 1, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
15 | fzossfz 13059 | . . . . . 6 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)) | |
16 | 15, 5 | sseldi 3968 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
17 | pfxres 14044 | . . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) | |
18 | 14, 16, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) |
19 | 11, 18 | syl5eq 2871 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ↾ (0..^𝑁))) |
20 | 11 | fveq2i 6676 | . . . . 5 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁)) |
21 | elfzofz 13056 | . . . . . . 7 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ (0...(♯‘𝐹))) | |
22 | 5, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
23 | pfxlen 14048 | . . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁) | |
24 | 14, 22, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁) |
25 | 20, 24 | syl5eq 2871 | . . . 4 ⊢ (𝜑 → (♯‘𝐻) = 𝑁) |
26 | 25 | oveq2d 7175 | . . 3 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁)) |
27 | wrdf 13869 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
28 | fimass 6558 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) | |
29 | 13, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) |
30 | 1, 12, 29 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) |
31 | ssdmres 5879 | . . . 4 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) | |
32 | 30, 31 | sylib 220 | . . 3 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) |
33 | 19, 26, 32 | f1oeq123d 6613 | . 2 ⊢ (𝜑 → (𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ↔ (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁)))) |
34 | 10, 33 | mpbird 259 | 1 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 class class class wbr 5069 dom cdm 5558 ↾ cres 5560 “ cima 5561 ⟶wf 6354 –1-1→wf1 6355 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 0cc0 10540 ℤ≥cuz 12246 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 prefix cpfx 14035 Vtxcvtx 26784 iEdgciedg 26785 Walkscwlks 27381 Trailsctrls 27475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
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 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-substr 14006 df-pfx 14036 df-wlks 27384 df-trls 27477 |
This theorem is referenced by: trlres 27485 eupthres 27997 |
Copyright terms: Public domain | W3C validator |