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Theorem efginvrel2 19597

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 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o))
2		fviss 6968	. . . 4 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
3	1, 2	eqsstri 4016	. . 3 ⊢ 𝑊 ⊆ Word (𝐼 × 2_o)
4	3	sseli 3978	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2_o))
5		id 22	. . . . . 6 ⊢ (𝑐 = ∅ → 𝑐 = ∅)
6		fveq2 6891	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7		rev0 14716	. . . . . . . . 9 ⊢ (reverse‘∅) = ∅
8	6, 7	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = ∅)
9	8	coeq2d 5862	. . . . . . 7 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10		co02 6259	. . . . . . 7 ⊢ (𝑀 ∘ ∅) = ∅
11	9, 10	eqtrdi 2788	. . . . . 6 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
12	5, 11	oveq12d 7429	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
13	12	breq1d 5158	. . . 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 6891	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
17	16	coeq2d 5862	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
18	15, 17	oveq12d 7429	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
19	18	breq1d 5158	. . . 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 6891	. . . . . . 7 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
23	22	coeq2d 5862	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
24	21, 23	oveq12d 7429	. . . . 5 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
25	24	breq1d 5158	. . . 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 6891	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
29	28	coeq2d 5862	. . . . . 6 ⊢ (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
30	27, 29	oveq12d 7429	. . . . 5 ⊢ (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
31	30	breq1d 5158	. . . 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 14542	. . . 4 ⊢ (∅ ++ ∅) = ∅
34		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~_FG ‘𝐼)
35	1, 34	efger 19588	. . . . . 6 ⊢ ∼ Er 𝑊
36	35	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
37		wrd0 14491	. . . . . 6 ⊢ ∅ ∈ Word (𝐼 × 2_o)
38	1	efgrcl 19585	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))
39	38	simprd 496	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2_o))
40	37, 39	eleqtrrid 2840	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∅ ∈ 𝑊)
41	36, 40	erref 8725	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∅ ∼ ∅)
42	33, 41	eqbrtrid 5183	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)
43	35	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ∼ Er 𝑊)
44		simprl 769	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 ∈ Word (𝐼 × 2_o))
45		revcl 14713	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2_o) → (reverse‘𝑎) ∈ Word (𝐼 × 2_o))
46	45	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (reverse‘𝑎) ∈ Word (𝐼 × 2_o))
47		efgval2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
48	47	efgmf 19583	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)
49		wrdco 14784	. . . . . . . . . . 11 ⊢ (((reverse‘𝑎) ∈ Word (𝐼 × 2_o) ∧ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o))
50	46, 48, 49	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o))
51		ccatcl 14526	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2_o))
52	44, 50, 51	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2_o))
53	39	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑊 = Word (𝐼 × 2_o))
54	52, 53	eleqtrrd 2836	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55		lencl 14485	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2_o) → (♯‘𝑎) ∈ ℕ₀)
56	55	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ ℕ₀)
57		nn0uz 12866	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
58	56, 57	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ (ℤ_≥‘0))
59		ccatlen 14527	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
60	44, 50, 59	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
61	56	nn0zd 12586	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ ℤ)
62	61	uzidd 12840	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ (ℤ_≥‘(♯‘𝑎)))
63		lencl 14485	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ₀)
64	50, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ₀)
65		uzaddcl 12890	. . . . . . . . . . . . . 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 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(♯‘𝑎)))
67	60, 66	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(♯‘𝑎)))
68		elfzuzb 13497	. . . . . . . . . . . 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 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70		simprr 771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑏 ∈ (𝐼 × 2_o))
71		efgval2.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
72	1, 34, 47, 71	efgtval 19593	. . . . . . . . . . 11 ⊢ (((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2_o)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
73	54, 69, 70, 72	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘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 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ∅ ∈ Word (𝐼 × 2_o))
75	48	ffvelcdmi 7085	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐼 × 2_o) → (𝑀‘𝑏) ∈ (𝐼 × 2_o))
76	75	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀‘𝑏) ∈ (𝐼 × 2_o))
77	70, 76	s2cld 14824	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_o))
78		ccatrid 14539	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2_o) → (𝑎 ++ ∅) = 𝑎)
79	78	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ ∅) = 𝑎)
80	79	eqcomd 2738	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 = (𝑎 ++ ∅))
81	80	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82		eqidd 2733	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) = (♯‘𝑎))
83		hash0 14329	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
84	83	oveq2i 7422	. . . . . . . . . . . 12 ⊢ ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
85	56	nn0cnd 12536	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) ∈ ℂ)
86	85	addridd 11416	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
87	84, 86	eqtr2id 2785	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
88	44, 74, 50, 77, 81, 82, 87	splval2 14709	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
89	70	s1cld 14555	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_o))
90		revccat 14718	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
91	44, 89, 90	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92		revs1 14717	. . . . . . . . . . . . . . . 16 ⊢ (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
93	92	oveq1i 7421	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
94	91, 93	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
95	94	coeq2d 5862	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))))
96	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o))
97		ccatco 14788	. . . . . . . . . . . . . 14 ⊢ ((⟨“𝑏”⟩ ∈ Word (𝐼 × 2_o) ∧ (reverse‘𝑎) ∈ Word (𝐼 × 2_o) ∧ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
98	89, 46, 96, 97	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99		s1co 14786	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (𝐼 × 2_o) ∧ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
100	70, 48, 99	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
101	100	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
102	95, 98, 101	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103	102	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
104		ccatcl 14526	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2_o))
105	44, 89, 104	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2_o))
106	76	s1cld 14555	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_o))
107		ccatass 14540	. . . . . . . . . . . 12 ⊢ (((𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_o)) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
108	105, 106, 50, 107	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
109		ccatass 14540	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_o)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
110	44, 89, 106, 109	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
111		df-s2 14801	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑏(𝑀‘𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)
112	111	oveq2i 7422	. . . . . . . . . . . . 13 ⊢ (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩))
113	110, 112	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩))
114	113	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
115	103, 108, 114	3eqtr2rd 2779	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
116	73, 88, 115	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
117	1, 34, 47, 71	efgtf 19592	. . . . . . . . . . . . 13 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → ((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = (𝑚 ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))), 𝑢 ∈ (𝐼 × 2_o) ↦ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ∧ (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_o))⟶𝑊))
118	117	simprd 496	. . . . . . . . . . . 12 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_o))⟶𝑊)
119	54, 118	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_o))⟶𝑊)
120	119	ffnd 6718	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_o)))
121		fnovrn 7584	. . . . . . . . . 10 ⊢ (((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_o)) ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2_o)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
122	120, 69, 70, 121	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
123	116, 122	eqeltrrd 2834	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
124	1, 34, 47, 71	efgi2 19595	. . . . . . . 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 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
126	43, 125	ersym 8717	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
127	43	ertr 8720	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∧ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
128	126, 127	mpand 693	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
129	128	expcom 414	. . . 4 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o)) → (𝐴 ∈ 𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
130	129	a2d 29	. . 3 ⊢ ((𝑎 ∈ Word (𝐼 × 2_o) ∧ 𝑏 ∈ (𝐼 × 2_o)) → ((𝐴 ∈ 𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅) → (𝐴 ∈ 𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
131	14, 20, 26, 32, 42, 130	wrdind 14674	. 2 ⊢ (𝐴 ∈ Word (𝐼 × 2_o) → (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
132	4, 131	mpcom 38	1 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3945 ∅c0 4322 ⟨cop 4634 ⟨cotp 4636 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 × cxp 5674 ran crn 5677 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 1_oc1o 8461 2_oc2o 8462 Er wer 8702 0cc0 11112 + caddc 11115 ℕ₀cn0 12474 ℤ_≥cuz 12824 ...cfz 13486 ♯chash 14292 Word cword 14466 ++ cconcat 14522 ⟨“cs1 14547 splice csplice 14701 reversecreverse 14710 ⟨“cs2 14794 ~_FG cefg 19576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-ec 8707 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 df-hash 14293 df-word 14467 df-lsw 14515 df-concat 14523 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14702 df-reverse 14711 df-s2 14801 df-efg 19579

This theorem is referenced by: efginvrel1 19598 frgpinv 19634

W3C validator