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Mirrors > Home > MPE Home > Th. List > efgrelex | Structured version Visualization version GIF version |
Description: If two words π΄, π΅ are related under the free group equivalence, then there exist two extension sequences π, π such that π ends at π΄, π ends at π΅, and π and π΅ have the same starting point. (Contributed by Mario Carneiro, 1-Oct-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))) |
Ref | Expression |
---|---|
efgrelex | β’ (π΄ βΌ π΅ β βπ β (β‘π β {π΄})βπ β (β‘π β {π΅})(πβ0) = (πβ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . 4 β’ π = ( I βWord (πΌ Γ 2o)) | |
2 | efgval.r | . . . 4 β’ βΌ = ( ~FG βπΌ) | |
3 | efgval2.m | . . . 4 β’ π = (π¦ β πΌ, π§ β 2o β¦ β¨π¦, (1o β π§)β©) | |
4 | efgval2.t | . . . 4 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 | . . . 4 β’ π· = (π β βͺ π₯ β π ran (πβπ₯)) | |
6 | efgred.s | . . . 4 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 | eqid 2728 | . . . 4 β’ {β¨π, πβ© β£ βπ β (β‘π β {π})βπ β (β‘π β {π})(πβ0) = (πβ0)} = {β¨π, πβ© β£ βπ β (β‘π β {π})βπ β (β‘π β {π})(πβ0) = (πβ0)} | |
8 | 1, 2, 3, 4, 5, 6, 7 | efgrelexlemb 19719 | . . 3 β’ βΌ β {β¨π, πβ© β£ βπ β (β‘π β {π})βπ β (β‘π β {π})(πβ0) = (πβ0)} |
9 | 8 | ssbri 5197 | . 2 β’ (π΄ βΌ π΅ β π΄{β¨π, πβ© β£ βπ β (β‘π β {π})βπ β (β‘π β {π})(πβ0) = (πβ0)}π΅) |
10 | 1, 2, 3, 4, 5, 6, 7 | efgrelexlema 19718 | . 2 β’ (π΄{β¨π, πβ© β£ βπ β (β‘π β {π})βπ β (β‘π β {π})(πβ0) = (πβ0)}π΅ β βπ β (β‘π β {π΄})βπ β (β‘π β {π΅})(πβ0) = (πβ0)) |
11 | 9, 10 | sylib 217 | 1 β’ (π΄ βΌ π΅ β βπ β (β‘π β {π΄})βπ β (β‘π β {π΅})(πβ0) = (πβ0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 {crab 3430 β cdif 3946 β c0 4326 {csn 4632 β¨cop 4638 β¨cotp 4640 βͺ ciun 5000 class class class wbr 5152 {copab 5214 β¦ cmpt 5235 I cid 5579 Γ cxp 5680 β‘ccnv 5681 ran crn 5683 β cima 5685 βcfv 6553 (class class class)co 7426 β cmpo 7428 1oc1o 8488 2oc2o 8489 0cc0 11148 1c1 11149 β cmin 11484 ...cfz 13526 ..^cfzo 13669 β―chash 14331 Word cword 14506 splice csplice 14741 β¨βcs2 14834 ~FG cefg 19675 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-ec 8735 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-rp 13017 df-fz 13527 df-fzo 13670 df-hash 14332 df-word 14507 df-concat 14563 df-s1 14588 df-substr 14633 df-pfx 14663 df-splice 14742 df-s2 14841 df-efg 19678 |
This theorem is referenced by: efgredeu 19721 |
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