Theorem efgcpbllema 19409

Description: Lemma for efgrelex 19406. Define an auxiliary equivalence relation 𝐿 such that 𝐴𝐿𝐵 if there are sequences from 𝐴 to 𝐵 passing through the same reduced word. (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)))
efgcpbllem.1	⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}

Assertion

Ref	Expression
efgcpbllema	⊢ (𝑋𝐿𝑌 ↔ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))

Distinct variable groups:   𝑖,𝑗,𝐴   𝑦,𝑧   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧   𝑖,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑗   𝑖,𝑘,𝑇,𝑗,𝑚,𝑡,𝑥   𝑖,𝑋,𝑗   𝑦,𝑖,𝑧,𝑊,𝑗   𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥   ∼ ,𝑖,𝑗,𝑚,𝑡,𝑥,𝑦,𝑧   𝐵,𝑖,𝑗   𝑆,𝑖,𝑗   𝑖,𝑌,𝑗   𝑖,𝐼,𝑗,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   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)   𝐼(𝑘)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑖,𝑗,𝑘,𝑚,𝑛)   𝑀(𝑦,𝑧,𝑘)   𝑋(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑌(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)

Proof of Theorem efgcpbllema

Step	Hyp	Ref	Expression
1		oveq2 7315	. . . . 5 ⊢ (𝑖 = 𝑋 → (𝐴 ++ 𝑖) = (𝐴 ++ 𝑋))
2	1	oveq1d 7322	. . . 4 ⊢ (𝑖 = 𝑋 → ((𝐴 ++ 𝑖) ++ 𝐵) = ((𝐴 ++ 𝑋) ++ 𝐵))
3		oveq2 7315	. . . . 5 ⊢ (𝑗 = 𝑌 → (𝐴 ++ 𝑗) = (𝐴 ++ 𝑌))
4	3	oveq1d 7322	. . . 4 ⊢ (𝑗 = 𝑌 → ((𝐴 ++ 𝑗) ++ 𝐵) = ((𝐴 ++ 𝑌) ++ 𝐵))
5	2, 4	breqan12d 5097	. . 3 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑌) → (((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵) ↔ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))
6		efgcpbllem.1	. . . 4 ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}
7		vex 3441	. . . . . . 7 ⊢ 𝑖 ∈ V
8		vex 3441	. . . . . . 7 ⊢ 𝑗 ∈ V
9	7, 8	prss 4759	. . . . . 6 ⊢ ((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ↔ {𝑖, 𝑗} ⊆ 𝑊)
10	9	anbi1i 625	. . . . 5 ⊢ (((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵)) ↔ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵)))
11	10	opabbii 5148	. . . 4 ⊢ {⟨𝑖, 𝑗⟩ ∣ ((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}
12	6, 11	eqtr4i 2767	. . 3 ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}
13	5, 12	brab2a 5691	. 2 ⊢ (𝑋𝐿𝑌 ↔ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))
14		df-3an 1089	. 2 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)) ↔ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))
15	13, 14	bitr4i 278	1 ⊢ (𝑋𝐿𝑌 ↔ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∀wral 3062  {crab 3303   ∖ cdif 3889   ⊆ wss 3892  ∅c0 4262  {csn 4565  {cpr 4567  ⟨cop 4571  ⟨cotp 4573  ∪ ciun 4931   class class class wbr 5081  {copab 5143   ↦ cmpt 5164   I cid 5499   × cxp 5598  ran crn 5601  ‘cfv 6458  (class class class)co 7307   ∈ cmpo 7309  1_oc1o 8321  2_oc2o 8322  0cc0 10921  1c1 10922   − cmin 11255  ...cfz 13289  ..^cfzo 13432  ♯chash 14094  Word cword 14266   ++ cconcat 14322   splice csplice 14511  ⟨“cs2 14603   ~_FG cefg 19361

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-xp 5606  df-iota 6410  df-fv 6466  df-ov 7310

This theorem is referenced by:  efgcpbllemb  19410  efgcpbl  19411