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Mirrors > Home > MPE Home > Th. List > trlreslem | Structured version Visualization version GIF version |
Description: Lemma for trlres 29558. Formerly part of proof of eupthres 30069. (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 | β’ π = (VtxβπΊ) |
trlres.i | β’ πΌ = (iEdgβπΊ) |
trlres.d | β’ (π β πΉ(TrailsβπΊ)π) |
trlres.n | β’ (π β π β (0..^(β―βπΉ))) |
trlres.h | β’ π» = (πΉ prefix π) |
Ref | Expression |
---|---|
trlreslem | β’ (π β π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlres.d | . . . 4 β’ (π β πΉ(TrailsβπΊ)π) | |
2 | trlres.i | . . . . 5 β’ πΌ = (iEdgβπΊ) | |
3 | 2 | trlf1 29556 | . . . 4 β’ (πΉ(TrailsβπΊ)π β πΉ:(0..^(β―βπΉ))β1-1βdom πΌ) |
4 | 1, 3 | syl 17 | . . 3 β’ (π β πΉ:(0..^(β―βπΉ))β1-1βdom πΌ) |
5 | trlres.n | . . . 4 β’ (π β π β (0..^(β―βπΉ))) | |
6 | elfzouz2 13679 | . . . 4 β’ (π β (0..^(β―βπΉ)) β (β―βπΉ) β (β€β₯βπ)) | |
7 | fzoss2 13692 | . . . 4 β’ ((β―βπΉ) β (β€β₯βπ) β (0..^π) β (0..^(β―βπΉ))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 β’ (π β (0..^π) β (0..^(β―βπΉ))) |
9 | f1ores 6848 | . . 3 β’ ((πΉ:(0..^(β―βπΉ))β1-1βdom πΌ β§ (0..^π) β (0..^(β―βπΉ))) β (πΉ βΎ (0..^π)):(0..^π)β1-1-ontoβ(πΉ β (0..^π))) | |
10 | 4, 8, 9 | syl2anc 582 | . 2 β’ (π β (πΉ βΎ (0..^π)):(0..^π)β1-1-ontoβ(πΉ β (0..^π))) |
11 | trlres.h | . . . 4 β’ π» = (πΉ prefix π) | |
12 | trliswlk 29555 | . . . . . 6 β’ (πΉ(TrailsβπΊ)π β πΉ(WalksβπΊ)π) | |
13 | 2 | wlkf 29472 | . . . . . 6 β’ (πΉ(WalksβπΊ)π β πΉ β Word dom πΌ) |
14 | 1, 12, 13 | 3syl 18 | . . . . 5 β’ (π β πΉ β Word dom πΌ) |
15 | fzossfz 13683 | . . . . . 6 β’ (0..^(β―βπΉ)) β (0...(β―βπΉ)) | |
16 | 15, 5 | sselid 3970 | . . . . 5 β’ (π β π β (0...(β―βπΉ))) |
17 | pfxres 14661 | . . . . 5 β’ ((πΉ β Word dom πΌ β§ π β (0...(β―βπΉ))) β (πΉ prefix π) = (πΉ βΎ (0..^π))) | |
18 | 14, 16, 17 | syl2anc 582 | . . . 4 β’ (π β (πΉ prefix π) = (πΉ βΎ (0..^π))) |
19 | 11, 18 | eqtrid 2777 | . . 3 β’ (π β π» = (πΉ βΎ (0..^π))) |
20 | 11 | fveq2i 6895 | . . . . 5 β’ (β―βπ») = (β―β(πΉ prefix π)) |
21 | elfzofz 13680 | . . . . . . 7 β’ (π β (0..^(β―βπΉ)) β π β (0...(β―βπΉ))) | |
22 | 5, 21 | syl 17 | . . . . . 6 β’ (π β π β (0...(β―βπΉ))) |
23 | pfxlen 14665 | . . . . . 6 β’ ((πΉ β Word dom πΌ β§ π β (0...(β―βπΉ))) β (β―β(πΉ prefix π)) = π) | |
24 | 14, 22, 23 | syl2anc 582 | . . . . 5 β’ (π β (β―β(πΉ prefix π)) = π) |
25 | 20, 24 | eqtrid 2777 | . . . 4 β’ (π β (β―βπ») = π) |
26 | 25 | oveq2d 7432 | . . 3 β’ (π β (0..^(β―βπ»)) = (0..^π)) |
27 | wrdf 14501 | . . . . . 6 β’ (πΉ β Word dom πΌ β πΉ:(0..^(β―βπΉ))βΆdom πΌ) | |
28 | fimass 6738 | . . . . . 6 β’ (πΉ:(0..^(β―βπΉ))βΆdom πΌ β (πΉ β (0..^π)) β dom πΌ) | |
29 | 13, 27, 28 | 3syl 18 | . . . . 5 β’ (πΉ(WalksβπΊ)π β (πΉ β (0..^π)) β dom πΌ) |
30 | 1, 12, 29 | 3syl 18 | . . . 4 β’ (π β (πΉ β (0..^π)) β dom πΌ) |
31 | ssdmres 6012 | . . . 4 β’ ((πΉ β (0..^π)) β dom πΌ β dom (πΌ βΎ (πΉ β (0..^π))) = (πΉ β (0..^π))) | |
32 | 30, 31 | sylib 217 | . . 3 β’ (π β dom (πΌ βΎ (πΉ β (0..^π))) = (πΉ β (0..^π))) |
33 | 19, 26, 32 | f1oeq123d 6828 | . 2 β’ (π β (π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π))) β (πΉ βΎ (0..^π)):(0..^π)β1-1-ontoβ(πΉ β (0..^π)))) |
34 | 10, 33 | mpbird 256 | 1 β’ (π β π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 class class class wbr 5143 dom cdm 5672 βΎ cres 5674 β cima 5675 βΆwf 6539 β1-1βwf1 6540 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7416 0cc0 11138 β€β₯cuz 12852 ...cfz 13516 ..^cfzo 13659 β―chash 14321 Word cword 14496 prefix cpfx 14652 Vtxcvtx 28853 iEdgciedg 28854 Walkscwlks 29454 Trailsctrls 29548 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-substr 14623 df-pfx 14653 df-wlks 29457 df-trls 29550 |
This theorem is referenced by: trlres 29558 eupthres 30069 |
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