Theorem efgrelexlema 19767

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	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ 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

Step	Hyp	Ref	Expression
1		efgrelexlem.1	. . 3 ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}
2	1	bropaex12 5777	. 2 ⊢ (𝐴𝐿𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		n0i 4340	. . . . . 6 ⊢ (𝑎 ∈ (◡𝑆 “ {𝐴}) → ¬ (◡𝑆 “ {𝐴}) = ∅)
4		snprc 4717	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5		imaeq2 6074	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (◡𝑆 “ {𝐴}) = (◡𝑆 “ ∅))
6	4, 5	sylbi 217	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (◡𝑆 “ {𝐴}) = (◡𝑆 “ ∅))
7		ima0 6095	. . . . . . 7 ⊢ (◡𝑆 “ ∅) = ∅
8	6, 7	eqtrdi 2793	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (◡𝑆 “ {𝐴}) = ∅)
9	3, 8	nsyl2 141	. . . . 5 ⊢ (𝑎 ∈ (◡𝑆 “ {𝐴}) → 𝐴 ∈ V)
10		n0i 4340	. . . . . 6 ⊢ (𝑏 ∈ (◡𝑆 “ {𝐵}) → ¬ (◡𝑆 “ {𝐵}) = ∅)
11		snprc 4717	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
12		imaeq2 6074	. . . . . . . 8 ⊢ ({𝐵} = ∅ → (◡𝑆 “ {𝐵}) = (◡𝑆 “ ∅))
13	11, 12	sylbi 217	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (◡𝑆 “ {𝐵}) = (◡𝑆 “ ∅))
14	13, 7	eqtrdi 2793	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → (◡𝑆 “ {𝐵}) = ∅)
15	10, 14	nsyl2 141	. . . . 5 ⊢ (𝑏 ∈ (◡𝑆 “ {𝐵}) → 𝐵 ∈ V)
16	9, 15	anim12i 613	. . . 4 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝐴}) ∧ 𝑏 ∈ (◡𝑆 “ {𝐵})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17	16	a1d 25	. . 3 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝐴}) ∧ 𝑏 ∈ (◡𝑆 “ {𝐵})) → ((𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
18	17	rexlimivv 3201	. 2 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19		fveq1 6905	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
20	19	eqeq1d 2739	. . . . 5 ⊢ (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
21		fveq1 6905	. . . . . 6 ⊢ (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
22	21	eqeq2d 2748	. . . . 5 ⊢ (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
23	20, 22	cbvrex2vw 3242	. . . 4 ⊢ (∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑖})∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0))
24		sneq 4636	. . . . . 6 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴})
25	24	imaeq2d 6078	. . . . 5 ⊢ (𝑖 = 𝐴 → (◡𝑆 “ {𝑖}) = (◡𝑆 “ {𝐴}))
26	25	rexeqdv 3327	. . . 4 ⊢ (𝑖 = 𝐴 → (∃𝑎 ∈ (◡𝑆 “ {𝑖})∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0)))
27	23, 26	bitrid 283	. . 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 4636	. . . . . 6 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵})
29	28	imaeq2d 6078	. . . . 5 ⊢ (𝑗 = 𝐵 → (◡𝑆 “ {𝑗}) = (◡𝑆 “ {𝐵}))
30	29	rexeqdv 3327	. . . 4 ⊢ (𝑗 = 𝐵 → (∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
31	30	rexbidv 3179	. . 3 ⊢ (𝑗 = 𝐵 → (∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
32	27, 31, 1	brabg 5544	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
33	2, 18, 32	pm5.21nii 378	1 ⊢ (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948  ∅c0 4333  {csn 4626  ⟨cop 4632  ⟨cotp 4634  ∪ ciun 4991   class class class wbr 5143  {copab 5205   ↦ cmpt 5225   I cid 5577   × cxp 5683  ^◡ccnv 5684  ran crn 5686   “ cima 5688  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1_oc1o 8499  2_oc2o 8500  0cc0 11155  1c1 11156   − cmin 11492  ...cfz 13547  ..^cfzo 13694  ♯chash 14369  Word cword 14552   splice csplice 14787  ⟨“cs2 14880   ~_FG cefg 19724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569

This theorem is referenced by:  efgrelexlemb  19768  efgrelex  19769