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Theorem efgrelexlema 18382
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 5349 . 2 (𝐴𝐿𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 n0i 4063 . . . . . 6 (𝑎 ∈ (𝑆 “ {𝐴}) → ¬ (𝑆 “ {𝐴}) = ∅)
4 snprc 4397 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
5 imaeq2 5620 . . . . . . . 8 ({𝐴} = ∅ → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
64, 5sylbi 207 . . . . . . 7 𝐴 ∈ V → (𝑆 “ {𝐴}) = (𝑆 “ ∅))
7 ima0 5639 . . . . . . 7 (𝑆 “ ∅) = ∅
86, 7syl6eq 2810 . . . . . 6 𝐴 ∈ V → (𝑆 “ {𝐴}) = ∅)
93, 8nsyl2 142 . . . . 5 (𝑎 ∈ (𝑆 “ {𝐴}) → 𝐴 ∈ V)
10 n0i 4063 . . . . . 6 (𝑏 ∈ (𝑆 “ {𝐵}) → ¬ (𝑆 “ {𝐵}) = ∅)
11 snprc 4397 . . . . . . . 8 𝐵 ∈ V ↔ {𝐵} = ∅)
12 imaeq2 5620 . . . . . . . 8 ({𝐵} = ∅ → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1311, 12sylbi 207 . . . . . . 7 𝐵 ∈ V → (𝑆 “ {𝐵}) = (𝑆 “ ∅))
1413, 7syl6eq 2810 . . . . . 6 𝐵 ∈ V → (𝑆 “ {𝐵}) = ∅)
1510, 14nsyl2 142 . . . . 5 (𝑏 ∈ (𝑆 “ {𝐵}) → 𝐵 ∈ V)
169, 15anim12i 591 . . . 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 3174 . 2 (∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19 fveq1 6352 . . . . . 6 (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
2019eqeq1d 2762 . . . . 5 (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
21 fveq1 6352 . . . . . 6 (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
2221eqeq2d 2770 . . . . 5 (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
2320, 22cbvrex2v 3319 . . . 4 (∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (𝑆 “ {𝑖})∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0))
24 sneq 4331 . . . . . 6 (𝑖 = 𝐴 → {𝑖} = {𝐴})
2524imaeq2d 5624 . . . . 5 (𝑖 = 𝐴 → (𝑆 “ {𝑖}) = (𝑆 “ {𝐴}))
2625rexeqdv 3284 . . . 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 272 . . 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 4331 . . . . . 6 (𝑗 = 𝐵 → {𝑗} = {𝐵})
2928imaeq2d 5624 . . . . 5 (𝑗 = 𝐵 → (𝑆 “ {𝑗}) = (𝑆 “ {𝐵}))
3029rexeqdv 3284 . . . 4 (𝑗 = 𝐵 → (∃𝑏 ∈ (𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
3130rexbidv 3190 . . 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 5144 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
332, 18, 32pm5.21nii 367 1 (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  c0 4058  {csn 4321  cop 4327  cotp 4329   ciun 4672   class class class wbr 4804  {copab 4864  cmpt 4881   I cid 5173   × cxp 5264  ccnv 5265  ran crn 5267  cima 5269  cfv 6049  (class class class)co 6814  cmpt2 6816  1𝑜c1o 7723  2𝑜c2o 7724  0cc0 10148  1c1 10149  cmin 10478  ...cfz 12539  ..^cfzo 12679  chash 13331  Word cword 13497   splice csplice 13502  ⟨“cs2 13806   ~FG cefg 18339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fv 6057
This theorem is referenced by:  efgrelexlemb  18383  efgrelex  18384
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