Theorem efgcpbllema 19974

Description: Lemma for efgrelex 19971. 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 7429	. . . . 5 ⊢ (𝑖 = 𝑋 → (𝐴 ++ 𝑖) = (𝐴 ++ 𝑋))
2	1	oveq1d 7436	. . . 4 ⊢ (𝑖 = 𝑋 → ((𝐴 ++ 𝑖) ++ 𝐵) = ((𝐴 ++ 𝑋) ++ 𝐵))
3		oveq2 7429	. . . . 5 ⊢ (𝑗 = 𝑌 → (𝐴 ++ 𝑗) = (𝐴 ++ 𝑌))
4	3	oveq1d 7436	. . . 4 ⊢ (𝑗 = 𝑌 → ((𝐴 ++ 𝑗) ++ 𝐵) = ((𝐴 ++ 𝑌) ++ 𝐵))
5	2, 4	breqan12d 5118	. . 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 3454	. . . . . . 7 ⊢ 𝑖 ∈ V
8		vex 3454	. . . . . . 7 ⊢ 𝑗 ∈ V
9	7, 8	prss 4775	. . . . . 6 ⊢ ((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ↔ {𝑖, 𝑗} ⊆ 𝑊)
10	9	anbi1i 636	. . . . 5 ⊢ (((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵)) ↔ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵)))
11	10	opabbii 5171	. . . 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 2786	. . 3 ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ((𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊) ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}
13	5, 12	brab2a 5743	. 2 ⊢ (𝑋𝐿𝑌 ↔ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊) ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))
14		df-3an 1105	. 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 281	1 ⊢ (𝑋𝐿𝑌 ↔ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  {csn 4584  {cpr 4586  ⟨cop 4834  ⟨cotp 4857  ∪ ciun 4950   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   I cid 5544   × cxp 5648  ran crn 5651  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  1_oc1o 8473  2_oc2o 8474  0cc0 11204  1c1 11205   − cmin 11545  ...cfz 13643  ..^cfzo 13792  ♯chash 14478  Word cword 14662   ++ cconcat 14719   splice csplice 14903  ⟨“cs2 14997   ~_FG cefg 19926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-iota 6494  df-fv 6547  df-ov 7424

This theorem is used by:  efgcpbllemb  19975  efgcpbl  19976