MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgrelexlema Structured version   Visualization version   GIF version

Theorem efgrelexlema 18804
Description: If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
efgred.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
efgred.s 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
efgrelexlem.1 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
Assertion
Ref Expression
efgrelexlema (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑖,𝑗,𝐴   𝑦,𝑎,𝑧,𝑏   𝐿,𝑎,𝑏   𝑛,𝑐,𝑡,𝑣,𝑤,𝑦,𝑧   𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏,𝑐,𝑖,𝑗   𝑘,𝑎,𝑇,𝑏,𝑐,𝑖,𝑗,𝑚,𝑡,𝑥   𝑊,𝑎,𝑏,𝑐   𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊,𝑖,𝑗   ,𝑎,𝑏,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑑,𝑖,𝑗   𝑆,𝑎,𝑏,𝑐,𝑑,𝑖,𝑗   𝐼,𝑎,𝑏,𝑐,𝑖,𝑗,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑐,𝑑,𝑖,𝑗,𝑚,𝑡
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛)   (𝑤,𝑣,𝑘,𝑛)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛,𝑑)   𝐼(𝑘,𝑑)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑖,𝑗,𝑘,𝑚,𝑛,𝑐,𝑑)   𝑀(𝑦,𝑧,𝑘,𝑑)

Proof of Theorem efgrelexlema
StepHypRef Expression
1 efgrelexlem.1 . . 3 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
21bropaex12 5635 . 2 (𝐴𝐿𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 n0i 4296 . . . . . 6 (𝑎 ∈ (𝑆 “ {𝐴}) → ¬ (𝑆 “ {𝐴}) = ∅)
4 snprc 4645 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
5 imaeq2 5918 . . . . . . . 8 ({𝐴} = ∅ → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
64, 5sylbi 218 . . . . . . 7 𝐴 ∈ V → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
7 ima0 5938 . . . . . . 7 (𝑆 “ ∅) = ∅
86, 7syl6eq 2869 . . . . . 6 𝐴 ∈ V → (𝑆 “ {𝐴}) = ∅)
93, 8nsyl2 143 . . . . 5 (𝑎 ∈ (𝑆 “ {𝐴}) → 𝐴 ∈ V)
10 n0i 4296 . . . . . 6 (𝑏 ∈ (𝑆 “ {𝐵}) → ¬ (𝑆 “ {𝐵}) = ∅)
11 snprc 4645 . . . . . . . 8 𝐵 ∈ V ↔ {𝐵} = ∅)
12 imaeq2 5918 . . . . . . . 8 ({𝐵} = ∅ → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1311, 12sylbi 218 . . . . . . 7 𝐵 ∈ V → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1413, 7syl6eq 2869 . . . . . 6 𝐵 ∈ V → (𝑆 “ {𝐵}) = ∅)
1510, 14nsyl2 143 . . . . 5 (𝑏 ∈ (𝑆 “ {𝐵}) → 𝐵 ∈ V)
169, 15anim12i 612 . . . 4 ((𝑎 ∈ (𝑆 “ {𝐴}) ∧ 𝑏 ∈ (𝑆 “ {𝐵})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1716a1d 25 . . 3 ((𝑎 ∈ (𝑆 “ {𝐴}) ∧ 𝑏 ∈ (𝑆 “ {𝐵})) → ((𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
1817rexlimivv 3289 . 2 (∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19 fveq1 6662 . . . . . 6 (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
2019eqeq1d 2820 . . . . 5 (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
21 fveq1 6662 . . . . . 6 (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
2221eqeq2d 2829 . . . . 5 (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
2320, 22cbvrex2vw 3460 . . . 4 (∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑖})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0))
24 sneq 4567 . . . . . 6 (𝑖 = 𝐴 → {𝑖} = {𝐴})
2524imaeq2d 5922 . . . . 5 (𝑖 = 𝐴 → (𝑆 “ {𝑖}) = (𝑆 “ {𝐴}))
2625rexeqdv 3414 . . . 4 (𝑖 = 𝐴 → (∃𝑎 ∈ (𝑆 “ {𝑖})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0)))
2723, 26syl5bb 284 . . 3 (𝑖 = 𝐴 → (∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0)))
28 sneq 4567 . . . . . 6 (𝑗 = 𝐵 → {𝑗} = {𝐵})
2928imaeq2d 5922 . . . . 5 (𝑗 = 𝐵 → (𝑆 “ {𝑗}) = (𝑆 “ {𝐵}))
3029rexeqdv 3414 . . . 4 (𝑗 = 𝐵 → (∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
3130rexbidv 3294 . . 3 (𝑗 = 𝐵 → (∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
3227, 31, 1brabg 5417 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
332, 18, 32pm5.21nii 380 1 (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  c0 4288  {csn 4557  cop 4563  cotp 4565   ciun 4910   class class class wbr 5057  {copab 5119  cmpt 5137   I cid 5452   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  cfv 6348  (class class class)co 7145  cmpo 7147  1oc1o 8084  2oc2o 8085  0cc0 10525  1c1 10526  cmin 10858  ...cfz 12880  ..^cfzo 13021  chash 13678  Word cword 13849   splice csplice 14099  ⟨“cs2 14191   ~FG cefg 18761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356
This theorem is referenced by:  efgrelexlemb  18805  efgrelex  18806
  Copyright terms: Public domain W3C validator