Theorem efginvrel2 19248

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 6827	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3951	. . 3 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4	3	sseli 3913	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o))
5		id 22	. . . . . 6 ⊢ (𝑐 = ∅ → 𝑐 = ∅)
6		fveq2 6756	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7		rev0 14405	. . . . . . . . 9 ⊢ (reverse‘∅) = ∅
8	6, 7	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = ∅)
9	8	coeq2d 5760	. . . . . . 7 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10		co02 6153	. . . . . . 7 ⊢ (𝑀 ∘ ∅) = ∅
11	9, 10	eqtrdi 2795	. . . . . 6 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
12	5, 11	oveq12d 7273	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
13	12	breq1d 5080	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (∅ ++ ∅) ∼ ∅))
14	13	imbi2d 340	. . 3 ⊢ (𝑐 = ∅ → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)))
15		id 22	. . . . . 6 ⊢ (𝑐 = 𝑎 → 𝑐 = 𝑎)
16		fveq2 6756	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
17	16	coeq2d 5760	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
18	15, 17	oveq12d 7273	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
19	18	breq1d 5080	. . . 4 ⊢ (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅))
20	19	imbi2d 340	. . 3 ⊢ (𝑐 = 𝑎 → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅)))
21		id 22	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → 𝑐 = (𝑎 ++ ⟨“𝑏”⟩))
22		fveq2 6756	. . . . . . 7 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
23	22	coeq2d 5760	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
24	21, 23	oveq12d 7273	. . . . 5 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
25	24	breq1d 5080	. . . 4 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
26	25	imbi2d 340	. . 3 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
27		id 22	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝑐 = 𝐴)
28		fveq2 6756	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
29	28	coeq2d 5760	. . . . . 6 ⊢ (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
30	27, 29	oveq12d 7273	. . . . 5 ⊢ (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
31	30	breq1d 5080	. . . 4 ⊢ (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
32	31	imbi2d 340	. . 3 ⊢ (𝑐 = 𝐴 → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)))
33		ccatidid 14223	. . . 4 ⊢ (∅ ++ ∅) = ∅
34		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
35	1, 34	efger 19239	. . . . . 6 ⊢ ∼ Er 𝑊
36	35	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
37		wrd0 14170	. . . . . 6 ⊢ ∅ ∈ Word (𝐼 × 2o)
38	1	efgrcl 19236	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
39	38	simprd 495	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
40	37, 39	eleqtrrid 2846	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∅ ∈ 𝑊)
41	36, 40	erref 8476	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∅ ∼ ∅)
42	33, 41	eqbrtrid 5105	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)
43	35	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∼ Er 𝑊)
44		simprl 767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ Word (𝐼 × 2o))
45		revcl 14402	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
46	45	ad2antrl 724	. . . . . . . . . . 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 19234	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
49		wrdco 14472	. . . . . . . . . . 11 ⊢ (((reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
50	46, 48, 49	sylancl 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
51		ccatcl 14205	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
52	44, 50, 51	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
53	39	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
54	52, 53	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55		lencl 14164	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
56	55	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℕ0)
57		nn0uz 12549	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
58	56, 57	eleqtrdi 2849	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘0))
59		ccatlen 14206	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
60	44, 50, 59	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
61	56	nn0zd 12353	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℤ)
62	61	uzidd 12527	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)))
63		lencl 14164	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
64	50, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
65		uzaddcl 12573	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
66	62, 64, 65	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
67	60, 66	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
68		elfzuzb 13179	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ≥‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎))))
69	58, 67, 68	sylanbrc 582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70		simprr 769	. . . . . . . . . . 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 19244	. . . . . . . . . . 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 1369	. . . . . . . . . 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	ffvelrni 6942	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
76	75	ad2antll 725	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
77	70, 76	s2cld 14512	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
78		ccatrid 14220	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (𝑎 ++ ∅) = 𝑎)
79	78	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ∅) = 𝑎)
80	79	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 ++ ∅))
81	80	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = (♯‘𝑎))
83		hash0 14010	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
84	83	oveq2i 7266	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
85	56	nn0cnd 12225	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℂ)
86	85	addid1d 11105	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
87	84, 86	eqtr2id 2792	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
88	44, 74, 50, 77, 81, 82, 87	splval2 14398	. . . . . . . . . 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 14236	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o))
90		revccat 14407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
91	44, 89, 90	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92		revs1 14406	. . . . . . . . . . . . . . . 16 ⊢ (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
93	92	oveq1i 7265	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
94	91, 93	eqtrdi 2795	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
95	94	coeq2d 5760	. . . . . . . . . . . . 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 14476	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99		s1co 14474	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
100	70, 48, 99	sylancl 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
101	100	oveq1d 7270	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103	102	oveq2d 7271	. . . . . . . . . . 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 14205	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
105	44, 89, 104	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
106	76	s1cld 14236	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
107		ccatass 14221	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 14221	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
111		df-s2 14489	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑏(𝑀‘𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)
112	111	oveq2i 7266	. . . . . . . . . . . . 13 ⊢ (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩))
113	110, 112	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩))
114	113	oveq1d 7270	. . . . . . . . . . 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 2785	. . . . . . . . . 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 2782	. . . . . . . . 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 19243	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊)
119	54, 118	syl 17	. . . . . . . . . . 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 6585	. . . . . . . . . 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 7425	. . . . . . . . . 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 1369	. . . . . . . . 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 2840	. . . . . . . 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 19246	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
126	43, 125	ersym 8468	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
127	43	ertr 8471	. . . . . 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 691	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
129	128	expcom 413	. . . 4 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o)) → (𝐴 ∈ 𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
130	129	a2d 29	. . 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 14363	. 2 ⊢ (𝐴 ∈ Word (𝐼 × 2o) → (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
132	4, 131	mpcom 38	1 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  ⟨cop 4564  ⟨cotp 4566   class class class wbr 5070   ↦ cmpt 5153   I cid 5479   × cxp 5578  ran crn 5581   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1_oc1o 8260  2_oc2o 8261   Er wer 8453  0cc0 10802   + caddc 10805  ℕ₀cn0 12163  ℤ_≥cuz 12511  ...cfz 13168  ♯chash 13972  Word cword 14145   ++ cconcat 14201  ⟨“cs1 14228   splice csplice 14390  reversecreverse 14399  ⟨“cs2 14482   ~_FG cefg 19227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-ec 8458  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-lsw 14194  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-splice 14391  df-reverse 14400  df-s2 14489  df-efg 19230

This theorem is referenced by:  efginvrel1  19249  frgpinv  19285