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Theorem efgrelexlema 19531
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 5723 . 2 (𝐴𝐿𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 n0i 4293 . . . . . 6 (𝑎 ∈ (𝑆 “ {𝐴}) → ¬ (𝑆 “ {𝐴}) = ∅)
4 snprc 4678 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
5 imaeq2 6009 . . . . . . . 8 ({𝐴} = ∅ → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
64, 5sylbi 216 . . . . . . 7 𝐴 ∈ V → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
7 ima0 6029 . . . . . . 7 (𝑆 “ ∅) = ∅
86, 7eqtrdi 2792 . . . . . 6 𝐴 ∈ V → (𝑆 “ {𝐴}) = ∅)
93, 8nsyl2 141 . . . . 5 (𝑎 ∈ (𝑆 “ {𝐴}) → 𝐴 ∈ V)
10 n0i 4293 . . . . . 6 (𝑏 ∈ (𝑆 “ {𝐵}) → ¬ (𝑆 “ {𝐵}) = ∅)
11 snprc 4678 . . . . . . . 8 𝐵 ∈ V ↔ {𝐵} = ∅)
12 imaeq2 6009 . . . . . . . 8 ({𝐵} = ∅ → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1311, 12sylbi 216 . . . . . . 7 𝐵 ∈ V → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1413, 7eqtrdi 2792 . . . . . 6 𝐵 ∈ V → (𝑆 “ {𝐵}) = ∅)
1510, 14nsyl2 141 . . . . 5 (𝑏 ∈ (𝑆 “ {𝐵}) → 𝐵 ∈ V)
169, 15anim12i 613 . . . 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 3196 . 2 (∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19 fveq1 6841 . . . . . 6 (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
2019eqeq1d 2738 . . . . 5 (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
21 fveq1 6841 . . . . . 6 (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
2221eqeq2d 2747 . . . . 5 (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
2320, 22cbvrex2vw 3228 . . . 4 (∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑖})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0))
24 sneq 4596 . . . . . 6 (𝑖 = 𝐴 → {𝑖} = {𝐴})
2524imaeq2d 6013 . . . . 5 (𝑖 = 𝐴 → (𝑆 “ {𝑖}) = (𝑆 “ {𝐴}))
2625rexeqdv 3314 . . . 4 (𝑖 = 𝐴 → (∃𝑎 ∈ (𝑆 “ {𝑖})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0)))
2723, 26bitrid 282 . . 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 4596 . . . . . 6 (𝑗 = 𝐵 → {𝑗} = {𝐵})
2928imaeq2d 6013 . . . . 5 (𝑗 = 𝐵 → (𝑆 “ {𝑗}) = (𝑆 “ {𝐵}))
3029rexeqdv 3314 . . . 4 (𝑗 = 𝐵 → (∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
3130rexbidv 3175 . . 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 5496 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
332, 18, 32pm5.21nii 379 1 (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  c0 4282  {csn 4586  cop 4592  cotp 4594   ciun 4954   class class class wbr 5105  {copab 5167  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  cfv 6496  (class class class)co 7357  cmpo 7359  1oc1o 8405  2oc2o 8406  0cc0 11051  1c1 11052  cmin 11385  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402   splice csplice 14637  ⟨“cs2 14730   ~FG cefg 19488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fv 6504
This theorem is referenced by:  efgrelexlemb  19532  efgrelex  19533
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