Theorem efginvrel2 19992

Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (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)”⟩⟩)))

Assertion

Ref	Expression
efginvrel2	⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

Distinct variable groups:   𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧   𝑛,𝑀,𝑣,𝑤   𝑛,𝑊,𝑣,𝑤,𝑦,𝑧   𝑦, ∼ ,𝑧   𝑛,𝐼,𝑣,𝑤,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑦, 𝑧, 𝑤, 𝑣, 𝑛)   ∼ (𝑤, 𝑣, 𝑛)   𝑇(𝑦, 𝑧, 𝑤, 𝑣, 𝑛)   𝑀(𝑦, 𝑧)

Proof of Theorem efginvrel2

Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		fviss 6993	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3959	. . 3 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4	3	sseli 3909	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o))
5		id 23	. . . . . 6 ⊢ (𝑐 = ∅ → 𝑐 = ∅)
6		fveq2 6916	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7		rev0 14955	. . . . . . . . 9 ⊢ (reverse‘∅) = ∅
8	6, 7	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = ∅)
9	8	coeq2d 5851	. . . . . . 7 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10		co02 6283	. . . . . . 7 ⊢ (𝑀 ∘ ∅) = ∅
11	9, 10	eqtrdi 2814	. . . . . 6 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
12	5, 11	oveq12d 7469	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
13	12	breq1d 5118	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (∅ ++ ∅) ∼ ∅))
14	13	imbi2d 343	. . 3 ⊢ (𝑐 = ∅ → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)))
15		id 23	. . . . . 6 ⊢ (𝑐 = 𝑎 → 𝑐 = 𝑎)
16		fveq2 6916	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
17	16	coeq2d 5851	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
18	15, 17	oveq12d 7469	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
19	18	breq1d 5118	. . . 4 ⊢ (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅))
20	19	imbi2d 343	. . 3 ⊢ (𝑐 = 𝑎 → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅)))
21		id 23	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → 𝑐 = (𝑎 ++ ⟨“𝑏”⟩))
22		fveq2 6916	. . . . . . 7 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
23	22	coeq2d 5851	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
24	21, 23	oveq12d 7469	. . . . 5 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
25	24	breq1d 5118	. . . 4 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
26	25	imbi2d 343	. . 3 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
27		id 23	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝑐 = 𝐴)
28		fveq2 6916	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
29	28	coeq2d 5851	. . . . . 6 ⊢ (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
30	27, 29	oveq12d 7469	. . . . 5 ⊢ (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
31	30	breq1d 5118	. . . 4 ⊢ (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
32	31	imbi2d 343	. . 3 ⊢ (𝑐 = 𝐴 → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)))
33		ccatidid 14778	. . . 4 ⊢ (∅ ++ ∅) = ∅
34		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
35	1, 34	efger 19983	. . . . . 6 ⊢ ∼ Er 𝑊
36	35	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
37		wrd0 14725	. . . . . 6 ⊢ ∅ ∈ Word (𝐼 × 2o)
38	1	efgrcl 19980	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
39	38	simprd 501	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
40	37, 39	eleqtrrid 2871	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∅ ∈ 𝑊)
41	36, 40	erref 8776	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∅ ∼ ∅)
42	33, 41	eqbrtrid 5146	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)
43	35	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∼ Er 𝑊)
44		simprl 783	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ Word (𝐼 × 2o))
45		revcl 14952	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
46	45	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
47		efgval2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
48	47	efgmf 19978	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
49		wrdco 15024	. . . . . . . . . . 11 ⊢ (((reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
50	46, 48, 49	sylancl 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
51		ccatcl 14760	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
52	44, 50, 51	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
53	39	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
54	52, 53	eleqtrrd 2867	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55		lencl 14719	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
56	55	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℕ0)
57		nn0uz 13043	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
58	56, 57	eleqtrdi 2874	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘0))
59		ccatlen 14761	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
60	44, 50, 59	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
61	56	nn0zd 12758	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℤ)
62	61	uzidd 13021	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)))
63		lencl 14719	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
64	50, 63	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
65		uzaddcl 13071	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
66	62, 64, 65	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
67	60, 66	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
68		elfzuzb 13690	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ≥‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎))))
69	58, 67, 68	sylanbrc 595	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70		simprr 785	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
71		efgval2.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
72	1, 34, 47, 71	efgtval 19988	. . . . . . . . . . 11 ⊢ (((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2o)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
73	54, 69, 70, 72	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
74	37	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
75	48	ffvelcdmi 7115	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
76	75	ad2antll 742	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
77	70, 76	s2cld 15064	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
78		ccatrid 14774	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (𝑎 ++ ∅) = 𝑎)
79	78	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ∅) = 𝑎)
80	79	eqcomd 2769	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 ++ ∅))
81	80	oveq1d 7466	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82		eqidd 2764	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = (♯‘𝑎))
83		hash0 14552	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
84	83	oveq2i 7462	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
85	56	nn0cnd 12709	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℂ)
86	85	addridd 11549	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
87	84, 86	eqtr2id 2811	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
88	44, 74, 50, 77, 81, 82, 87	splval2 14948	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
89	70	s1cld 14791	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o))
90		revccat 14957	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
91	44, 89, 90	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92		revs1 14956	. . . . . . . . . . . . . . . 16 ⊢ (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
93	92	oveq1i 7461	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
94	91, 93	eqtrdi 2814	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
95	94	coeq2d 5851	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))))
96	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
97		ccatco 15028	. . . . . . . . . . . . . 14 ⊢ ((⟨“𝑏”⟩ ∈ Word (𝐼 × 2o) ∧ (reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
98	89, 46, 96, 97	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99		s1co 15026	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
100	70, 48, 99	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
101	100	oveq1d 7466	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
102	95, 98, 101	3eqtrd 2802	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103	102	oveq2d 7467	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
104		ccatcl 14760	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
105	44, 89, 104	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
106	76	s1cld 14791	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
107		ccatass 14775	. . . . . . . . . . . 12 ⊢ (((𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o) ∧ ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
108	105, 106, 50, 107	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
109		ccatass 14775	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o) ∧ ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
110	44, 89, 106, 109	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
111		df-s2 15041	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑏(𝑀‘𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)
112	111	oveq2i 7462	. . . . . . . . . . . . 13 ⊢ (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩))
113	110, 112	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩))
114	113	oveq1d 7466	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
115	103, 108, 114	3eqtr2rd 2805	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
116	73, 88, 115	3eqtrd 2802	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
117	1, 34, 47, 71	efgtf 19987	. . . . . . . . . . . . 13 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → ((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = (𝑚 ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))), 𝑢 ∈ (𝐼 × 2o) ↦ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ∧ (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊))
118	117	simprd 501	. . . . . . . . . . . 12 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊)
119	54, 118	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊)
120	119	ffnd 6740	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o)))
121		fnovrn 7628	. . . . . . . . . 10 ⊢ (((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o)) ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2o)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
122	120, 69, 70, 121	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
123	116, 122	eqeltrrd 2865	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
124	1, 34, 47, 71	efgi2 19990	. . . . . . . 8 ⊢ (((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 ∧ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
125	54, 123, 124	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
126	43, 125	ersym 8768	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
127	43	ertr 8771	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∧ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
128	126, 127	mpand 708	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
129	128	expcom 419	. . . 4 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o)) → (𝐴 ∈ 𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
130	129	a2d 30	. . 3 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o)) → ((𝐴 ∈ 𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅) → (𝐴 ∈ 𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
131	14, 20, 26, 32, 42, 130	wrdind 14913	. 2 ⊢ (𝐴 ∈ Word (𝐼 × 2o) → (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
132	4, 131	mpcom 39	1 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  ⟨cop 4840  ⟨cotp 4863   class class class wbr 5108   ↦ cmpt 5192   I cid 5552   × cxp 5656  ran crn 5659   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1_oc1o 8507  2_oc2o 8508   Er wer 8752  0cc0 11239   + caddc 11242  ℕ₀cn0 12646  ℤ_≥cuz 13005  ...cfz 13679  ♯chash 14515  Word cword 14699   ++ cconcat 14756  ⟨“cs1 14783   splice csplice 14940  reversecreverse 14949  ⟨“cs2 15034   ~_FG cefg 19971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-ec 8757  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-lsw 14749  df-concat 14757  df-s1 14784  df-substr 14831  df-pfx 14863  df-splice 14941  df-reverse 14950  df-s2 15041  df-efg 19974

This theorem is used by:  efginvrel1  19993  frgpinv  20029