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| Mirrors > Home > MPE Home > Th. List > trlreslem | Structured version Visualization version GIF version | ||
| Description: Lemma for trlres 29718. Formerly part of proof of eupthres 30234. (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 29716 | . . . 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 13714 | . . . 4 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
| 7 | fzoss2 13727 | . . . 4 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
| 8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
| 9 | f1ores 6862 | . . 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 584 | . 2 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) |
| 11 | trlres.h | . . . 4 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
| 12 | trliswlk 29715 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
| 13 | 2 | wlkf 29632 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
| 14 | 1, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
| 15 | fzossfz 13718 | . . . . . 6 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)) | |
| 16 | 15, 5 | sselid 3981 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
| 17 | pfxres 14717 | . . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) | |
| 18 | 14, 16, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) |
| 19 | 11, 18 | eqtrid 2789 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ↾ (0..^𝑁))) |
| 20 | 11 | fveq2i 6909 | . . . . 5 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁)) |
| 21 | elfzofz 13715 | . . . . . . 7 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ (0...(♯‘𝐹))) | |
| 22 | 5, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
| 23 | pfxlen 14721 | . . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁) | |
| 24 | 14, 22, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁) |
| 25 | 20, 24 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → (♯‘𝐻) = 𝑁) |
| 26 | 25 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁)) |
| 27 | wrdf 14557 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
| 28 | fimass 6756 | . . . . . 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 6031 | . . . 4 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) | |
| 32 | 30, 31 | sylib 218 | . . 3 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) |
| 33 | 19, 26, 32 | f1oeq123d 6842 | . 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 257 | 1 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 “ cima 5688 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℤ≥cuz 12878 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 prefix cpfx 14708 Vtxcvtx 29013 iEdgciedg 29014 Walkscwlks 29614 Trailsctrls 29708 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-substr 14679 df-pfx 14709 df-wlks 29617 df-trls 29710 |
| This theorem is referenced by: trlres 29718 eupthres 30234 |
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