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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 19688 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 6891 | . . . . . . . . . 10 β’ ((πβπ΄) = (πβπ΅) β (β―β(πβπ΄)) = (β―β(πβπ΅))) | |
10 | 9 | breq2d 5155 | . . . . . . . . 9 β’ ((πβπ΄) = (πβπ΅) β ((β―β(πβπ)) < (β―β(πβπ΄)) β (β―β(πβπ)) < (β―β(πβπ΅)))) |
11 | 10 | imbi1d 340 | . . . . . . . 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 3209 | . . . . . . 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 2731 | . . . . 5 β’ (π β (πβπ΅) = (πβπ΄)) |
18 | efgredlem.5 | . . . . . 6 β’ (π β Β¬ (π΄β0) = (π΅β0)) | |
19 | eqcom 2732 | . . . . . 6 β’ ((π΄β0) = (π΅β0) β (π΅β0) = (π΄β0)) | |
20 | 18, 19 | sylnib 327 | . . . . 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 2732 | . . . . . 6 β’ ((π΄βπΎ) = (π΅βπΏ) β (π΅βπΏ) = (π΄βπΎ)) | |
31 | 29, 30 | sylnib 327 | . . . . 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 19702 | . . . 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 19702 | . . 3 β’ (π β (π β (β€β₯βπ) β (π΄β0) = (π΅β0))) |
35 | 24 | elfzelzd 13532 | . . . 4 β’ (π β π β β€) |
36 | 23 | elfzelzd 13532 | . . . 4 β’ (π β π β β€) |
37 | uztric 12874 | . . . 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 582 | . . 3 β’ (π β (π β (β€β₯βπ) β¨ π β (β€β₯βπ))) |
39 | 33, 34, 38 | mpjaod 858 | . 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 394 β¨ wo 845 = wceq 1533 β wcel 2098 βwral 3051 {crab 3419 β cdif 3937 β c0 4318 {csn 4624 β¨cop 4630 β¨cotp 4632 βͺ ciun 4991 class class class wbr 5143 β¦ cmpt 5226 I cid 5569 Γ cxp 5670 dom cdm 5672 ran crn 5673 βcfv 6542 (class class class)co 7415 β cmpo 7417 1oc1o 8476 2oc2o 8477 0cc0 11136 1c1 11137 < clt 11276 β cmin 11472 β€cz 12586 β€β₯cuz 12850 ...cfz 13514 ..^cfzo 13657 β―chash 14319 Word cword 14494 splice csplice 14729 β¨βcs2 14822 ~FG cefg 19663 |
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 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-ot 4633 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-splice 14730 df-s2 14829 |
This theorem is referenced by: efgredlem 19704 |
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