Theorem efginvrel2 19606

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 6900	. . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3982	. . 3 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4	3	sseli 3931	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o))
5		id 22	. . . . . 6 ⊢ (𝑐 = ∅ → 𝑐 = ∅)
6		fveq2 6822	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7		rev0 14670	. . . . . . . . 9 ⊢ (reverse‘∅) = ∅
8	6, 7	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = ∅)
9	8	coeq2d 5805	. . . . . . 7 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10		co02 6209	. . . . . . 7 ⊢ (𝑀 ∘ ∅) = ∅
11	9, 10	eqtrdi 2780	. . . . . 6 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
12	5, 11	oveq12d 7367	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
13	12	breq1d 5102	. . . 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 6822	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
17	16	coeq2d 5805	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
18	15, 17	oveq12d 7367	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
19	18	breq1d 5102	. . . 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 6822	. . . . . . 7 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
23	22	coeq2d 5805	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
24	21, 23	oveq12d 7367	. . . . 5 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
25	24	breq1d 5102	. . . 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 6822	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
29	28	coeq2d 5805	. . . . . 6 ⊢ (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
30	27, 29	oveq12d 7367	. . . . 5 ⊢ (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
31	30	breq1d 5102	. . . 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 14497	. . . 4 ⊢ (∅ ++ ∅) = ∅
34		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
35	1, 34	efger 19597	. . . . . 6 ⊢ ∼ Er 𝑊
36	35	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
37		wrd0 14446	. . . . . 6 ⊢ ∅ ∈ Word (𝐼 × 2o)
38	1	efgrcl 19594	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
39	38	simprd 495	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
40	37, 39	eleqtrrid 2835	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∅ ∈ 𝑊)
41	36, 40	erref 8645	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∅ ∼ ∅)
42	33, 41	eqbrtrid 5127	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)
43	35	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∼ Er 𝑊)
44		simprl 770	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ Word (𝐼 × 2o))
45		revcl 14667	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
46	45	ad2antrl 728	. . . . . . . . . . 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 19592	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
49		wrdco 14738	. . . . . . . . . . 11 ⊢ (((reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
50	46, 48, 49	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
51		ccatcl 14481	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
52	44, 50, 51	syl2anc 584	. . . . . . . . 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 2831	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55		lencl 14440	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
56	55	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℕ0)
57		nn0uz 12777	. . . . . . . . . . . . 13 ⊢ ℕ0 = (ℤ≥‘0)
58	56, 57	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘0))
59		ccatlen 14482	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
60	44, 50, 59	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
61	56	nn0zd 12497	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℤ)
62	61	uzidd 12751	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)))
63		lencl 14440	. . . . . . . . . . . . . . 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 12805	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑎) ∈ (ℤ≥‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
66	62, 64, 65	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
67	60, 66	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎)))
68		elfzuzb 13421	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ≥‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ≥‘(♯‘𝑎))))
69	58, 67, 68	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70		simprr 772	. . . . . . . . . . 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 19602	. . . . . . . . . . 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 1373	. . . . . . . . . 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 7017	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
76	75	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
77	70, 76	s2cld 14778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
78		ccatrid 14494	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2o) → (𝑎 ++ ∅) = 𝑎)
79	78	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ∅) = 𝑎)
80	79	eqcomd 2735	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 ++ ∅))
81	80	oveq1d 7364	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82		eqidd 2730	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = (♯‘𝑎))
83		hash0 14274	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
84	83	oveq2i 7360	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
85	56	nn0cnd 12447	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℂ)
86	85	addridd 11316	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
87	84, 86	eqtr2id 2777	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
88	44, 74, 50, 77, 81, 82, 87	splval2 14663	. . . . . . . . . 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 14510	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o))
90		revccat 14672	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
91	44, 89, 90	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92		revs1 14671	. . . . . . . . . . . . . . . 16 ⊢ (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
93	92	oveq1i 7359	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
94	91, 93	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
95	94	coeq2d 5805	. . . . . . . . . . . . 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 14742	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99		s1co 14740	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
100	70, 48, 99	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
101	100	oveq1d 7364	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103	102	oveq2d 7365	. . . . . . . . . . 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 14481	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
105	44, 89, 104	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
106	76	s1cld 14510	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
107		ccatass 14495	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 14495	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
111		df-s2 14755	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑏(𝑀‘𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)
112	111	oveq2i 7360	. . . . . . . . . . . . 13 ⊢ (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩))
113	110, 112	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩))
114	113	oveq1d 7364	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2768	. . . . . . . . 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 19601	. . . . . . . . . . . . 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 6653	. . . . . . . . . 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 7524	. . . . . . . . . 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 1373	. . . . . . . . 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 2829	. . . . . . . 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 19604	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
126	43, 125	ersym 8637	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
127	43	ertr 8640	. . . . . 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 695	. . . . 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 14628	. 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 1540   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900  ∅c0 4284  ⟨cop 4583  ⟨cotp 4585   class class class wbr 5092   ↦ cmpt 5173   I cid 5513   × cxp 5617  ran crn 5620   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1_oc1o 8381  2_oc2o 8382   Er wer 8622  0cc0 11009   + caddc 11012  ℕ₀cn0 12384  ℤ_≥cuz 12735  ...cfz 13410  ♯chash 14237  Word cword 14420   ++ cconcat 14477  ⟨“cs1 14502   splice csplice 14655  reversecreverse 14664  ⟨“cs2 14748   ~_FG cefg 19585

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-ec 8627  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-hash 14238  df-word 14421  df-lsw 14470  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-reverse 14665  df-s2 14755  df-efg 19588

This theorem is referenced by:  efginvrel1  19607  frgpinv  19643