Theorem efgrelexlema 19807

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 5742	. 2 ⊢ (𝐴𝐿𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		n0i 4295	. . . . . 6 ⊢ (𝑎 ∈ (◡𝑆 “ {𝐴}) → ¬ (◡𝑆 “ {𝐴}) = ∅)
4		snprc 4679	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5		imaeq2 6048	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (◡𝑆 “ {𝐴}) = (◡𝑆 “ ∅))
6	4, 5	sylbi 220	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (◡𝑆 “ {𝐴}) = (◡𝑆 “ ∅))
7		ima0 6069	. . . . . . 7 ⊢ (◡𝑆 “ ∅) = ∅
8	6, 7	eqtrdi 2816	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (◡𝑆 “ {𝐴}) = ∅)
9	3, 8	nsyl2 142	. . . . 5 ⊢ (𝑎 ∈ (◡𝑆 “ {𝐴}) → 𝐴 ∈ V)
10		n0i 4295	. . . . . 6 ⊢ (𝑏 ∈ (◡𝑆 “ {𝐵}) → ¬ (◡𝑆 “ {𝐵}) = ∅)
11		snprc 4679	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
12		imaeq2 6048	. . . . . . . 8 ⊢ ({𝐵} = ∅ → (◡𝑆 “ {𝐵}) = (◡𝑆 “ ∅))
13	11, 12	sylbi 220	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (◡𝑆 “ {𝐵}) = (◡𝑆 “ ∅))
14	13, 7	eqtrdi 2816	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → (◡𝑆 “ {𝐵}) = ∅)
15	10, 14	nsyl2 142	. . . . 5 ⊢ (𝑏 ∈ (◡𝑆 “ {𝐵}) → 𝐵 ∈ V)
16	9, 15	anim12i 624	. . . 4 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝐴}) ∧ 𝑏 ∈ (◡𝑆 “ {𝐵})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17	16	a1d 26	. . 3 ⊢ ((𝑎 ∈ (◡𝑆 “ {𝐴}) ∧ 𝑏 ∈ (◡𝑆 “ {𝐵})) → ((𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
18	17	rexlimivv 3207	. 2 ⊢ (∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19		fveq1 6870	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
20	19	eqeq1d 2767	. . . . 5 ⊢ (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
21		fveq1 6870	. . . . . 6 ⊢ (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
22	21	eqeq2d 2776	. . . . 5 ⊢ (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
23	20, 22	cbvrex2vw 3248	. . . 4 ⊢ (∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0) ↔ ∃𝑎 ∈ (◡𝑆 “ {𝑖})∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0))
24		sneq 4595	. . . . . 6 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴})
25	24	imaeq2d 6052	. . . . 5 ⊢ (𝑖 = 𝐴 → (◡𝑆 “ {𝑖}) = (◡𝑆 “ {𝐴}))
26	25	rexeqdv 3324	. . . 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 286	. . 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 4595	. . . . . 6 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵})
29	28	imaeq2d 6052	. . . . 5 ⊢ (𝑗 = 𝐵 → (◡𝑆 “ {𝑗}) = (◡𝑆 “ {𝐵}))
30	29	rexeqdv 3324	. . . 4 ⊢ (𝑗 = 𝐵 → (∃𝑏 ∈ (◡𝑆 “ {𝑗})(𝑎‘0) = (𝑏‘0) ↔ ∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
31	30	rexbidv 3189	. . 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 5514	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)))
33	2, 18, 32	pm5.21nii 381	1 ⊢ (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904  ∅c0 4288  {csn 4585  ⟨cop 4591  ⟨cotp 4593  ∪ ciun 4951   class class class wbr 5104  {copab 5166   ↦ cmpt 5185   I cid 5545   × cxp 5649  ^◡ccnv 5650  ran crn 5652   “ cima 5654  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1_oc1o 8434  2_oc2o 8435  0cc0 11088  1c1 11089   − cmin 11429  ...cfz 13523  ..^cfzo 13670  ♯chash 14354  Word cword 14538   splice csplice 14774  ⟨“cs2 14866   ~_FG cefg 19764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fv 6533

This theorem is referenced by:  efgrelexlemb  19808  efgrelex  19809