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Mirrors > Home > MPE Home > Th. List > efgredlemb | Structured version Visualization version GIF version |
Description: The reduced word that forms the base of the sequence in efgsval 19651 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) |
Ref | Expression |
---|---|
efgval.w | β’ π = ( I βWord (πΌ Γ 2o)) |
efgval.r | β’ βΌ = ( ~FG βπΌ) |
efgval2.m | β’ π = (π¦ β πΌ, π§ β 2o β¦ β¨π¦, (1o β π§)β©) |
efgval2.t | β’ π = (π£ β π β¦ (π β (0...(β―βπ£)), π€ β (πΌ Γ 2o) β¦ (π£ splice β¨π, π, β¨βπ€(πβπ€)ββ©β©))) |
efgred.d | β’ π· = (π β βͺ π₯ β π ran (πβπ₯)) |
efgred.s | β’ π = (π β {π‘ β (Word π β {β }) β£ ((π‘β0) β π· β§ βπ β (1..^(β―βπ‘))(π‘βπ) β ran (πβ(π‘β(π β 1))))} β¦ (πβ((β―βπ) β 1))) |
efgredlem.1 | β’ (π β βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΄)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0)))) |
efgredlem.2 | β’ (π β π΄ β dom π) |
efgredlem.3 | β’ (π β π΅ β dom π) |
efgredlem.4 | β’ (π β (πβπ΄) = (πβπ΅)) |
efgredlem.5 | β’ (π β Β¬ (π΄β0) = (π΅β0)) |
efgredlemb.k | β’ πΎ = (((β―βπ΄) β 1) β 1) |
efgredlemb.l | β’ πΏ = (((β―βπ΅) β 1) β 1) |
efgredlemb.p | β’ (π β π β (0...(β―β(π΄βπΎ)))) |
efgredlemb.q | β’ (π β π β (0...(β―β(π΅βπΏ)))) |
efgredlemb.u | β’ (π β π β (πΌ Γ 2o)) |
efgredlemb.v | β’ (π β π β (πΌ Γ 2o)) |
efgredlemb.6 | β’ (π β (πβπ΄) = (π(πβ(π΄βπΎ))π)) |
efgredlemb.7 | β’ (π β (πβπ΅) = (π(πβ(π΅βπΏ))π)) |
efgredlemb.8 | β’ (π β Β¬ (π΄βπΎ) = (π΅βπΏ)) |
Ref | Expression |
---|---|
efgredlemb | β’ Β¬ π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . . 5 β’ π = ( I βWord (πΌ Γ 2o)) | |
2 | efgval.r | . . . . 5 β’ βΌ = ( ~FG βπΌ) | |
3 | efgval2.m | . . . . 5 β’ π = (π¦ β πΌ, π§ β 2o β¦ β¨π¦, (1o β π§)β©) | |
4 | efgval2.t | . . . . 5 β’ π = (π£ β π β¦ (π β (0...(β―βπ£)), π€ β (πΌ Γ 2o) β¦ (π£ splice β¨π, π, β¨βπ€(πβπ€)ββ©β©))) | |
5 | efgred.d | . . . . 5 β’ π· = (π β βͺ π₯ β π ran (πβπ₯)) | |
6 | efgred.s | . . . . 5 β’ π = (π β {π‘ β (Word π β {β }) β£ ((π‘β0) β π· β§ βπ β (1..^(β―βπ‘))(π‘βπ) β ran (πβ(π‘β(π β 1))))} β¦ (πβ((β―βπ) β 1))) | |
7 | efgredlem.1 | . . . . . 6 β’ (π β βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΄)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0)))) | |
8 | efgredlem.4 | . . . . . . 7 β’ (π β (πβπ΄) = (πβπ΅)) | |
9 | fveq2 6885 | . . . . . . . . . 10 β’ ((πβπ΄) = (πβπ΅) β (β―β(πβπ΄)) = (β―β(πβπ΅))) | |
10 | 9 | breq2d 5153 | . . . . . . . . 9 β’ ((πβπ΄) = (πβπ΅) β ((β―β(πβπ)) < (β―β(πβπ΄)) β (β―β(πβπ)) < (β―β(πβπ΅)))) |
11 | 10 | imbi1d 341 | . . . . . . . 8 β’ ((πβπ΄) = (πβπ΅) β (((β―β(πβπ)) < (β―β(πβπ΄)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))) β ((β―β(πβπ)) < (β―β(πβπ΅)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))))) |
12 | 11 | 2ralbidv 3212 | . . . . . . 7 β’ ((πβπ΄) = (πβπ΅) β (βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΄)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))) β βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΅)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))))) |
13 | 8, 12 | syl 17 | . . . . . 6 β’ (π β (βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΄)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))) β βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΅)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0))))) |
14 | 7, 13 | mpbid 231 | . . . . 5 β’ (π β βπ β dom πβπ β dom π((β―β(πβπ)) < (β―β(πβπ΅)) β ((πβπ) = (πβπ) β (πβ0) = (πβ0)))) |
15 | efgredlem.3 | . . . . 5 β’ (π β π΅ β dom π) | |
16 | efgredlem.2 | . . . . 5 β’ (π β π΄ β dom π) | |
17 | 8 | eqcomd 2732 | . . . . 5 β’ (π β (πβπ΅) = (πβπ΄)) |
18 | efgredlem.5 | . . . . . 6 β’ (π β Β¬ (π΄β0) = (π΅β0)) | |
19 | eqcom 2733 | . . . . . 6 β’ ((π΄β0) = (π΅β0) β (π΅β0) = (π΄β0)) | |
20 | 18, 19 | sylnib 328 | . . . . 5 β’ (π β Β¬ (π΅β0) = (π΄β0)) |
21 | efgredlemb.l | . . . . 5 β’ πΏ = (((β―βπ΅) β 1) β 1) | |
22 | efgredlemb.k | . . . . 5 β’ πΎ = (((β―βπ΄) β 1) β 1) | |
23 | efgredlemb.q | . . . . 5 β’ (π β π β (0...(β―β(π΅βπΏ)))) | |
24 | efgredlemb.p | . . . . 5 β’ (π β π β (0...(β―β(π΄βπΎ)))) | |
25 | efgredlemb.v | . . . . 5 β’ (π β π β (πΌ Γ 2o)) | |
26 | efgredlemb.u | . . . . 5 β’ (π β π β (πΌ Γ 2o)) | |
27 | efgredlemb.7 | . . . . 5 β’ (π β (πβπ΅) = (π(πβ(π΅βπΏ))π)) | |
28 | efgredlemb.6 | . . . . 5 β’ (π β (πβπ΄) = (π(πβ(π΄βπΎ))π)) | |
29 | efgredlemb.8 | . . . . . 6 β’ (π β Β¬ (π΄βπΎ) = (π΅βπΏ)) | |
30 | eqcom 2733 | . . . . . 6 β’ ((π΄βπΎ) = (π΅βπΏ) β (π΅βπΏ) = (π΄βπΎ)) | |
31 | 29, 30 | sylnib 328 | . . . . 5 β’ (π β Β¬ (π΅βπΏ) = (π΄βπΎ)) |
32 | 1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31 | efgredlemc 19665 | . . . 4 β’ (π β (π β (β€β₯βπ) β (π΅β0) = (π΄β0))) |
33 | 32, 19 | imbitrrdi 251 | . . 3 β’ (π β (π β (β€β₯βπ) β (π΄β0) = (π΅β0))) |
34 | 1, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29 | efgredlemc 19665 | . . 3 β’ (π β (π β (β€β₯βπ) β (π΄β0) = (π΅β0))) |
35 | 24 | elfzelzd 13508 | . . . 4 β’ (π β π β β€) |
36 | 23 | elfzelzd 13508 | . . . 4 β’ (π β π β β€) |
37 | uztric 12850 | . . . 4 β’ ((π β β€ β§ π β β€) β (π β (β€β₯βπ) β¨ π β (β€β₯βπ))) | |
38 | 35, 36, 37 | syl2anc 583 | . . 3 β’ (π β (π β (β€β₯βπ) β¨ π β (β€β₯βπ))) |
39 | 33, 34, 38 | mpjaod 857 | . 2 β’ (π β (π΄β0) = (π΅β0)) |
40 | 39, 18 | pm2.65i 193 | 1 β’ Β¬ π |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β¨ wo 844 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 β cdif 3940 β c0 4317 {csn 4623 β¨cop 4629 β¨cotp 4631 βͺ ciun 4990 class class class wbr 5141 β¦ cmpt 5224 I cid 5566 Γ cxp 5667 dom cdm 5669 ran crn 5670 βcfv 6537 (class class class)co 7405 β cmpo 7407 1oc1o 8460 2oc2o 8461 0cc0 11112 1c1 11113 < clt 11252 β cmin 11448 β€cz 12562 β€β₯cuz 12826 ...cfz 13490 ..^cfzo 13633 β―chash 14295 Word cword 14470 splice csplice 14705 β¨βcs2 14798 ~FG cefg 19626 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-splice 14706 df-s2 14805 |
This theorem is referenced by: efgredlem 19667 |
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