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Theorem efginvrel2 18140

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 6256	. . . 4 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
3	1, 2	eqsstri 3635	. . 3 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
4	3	sseli 3599	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2_𝑜))
5		id 22	. . . . . 6 ⊢ (𝑐 = ∅ → 𝑐 = ∅)
6		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7		rev0 13513	. . . . . . . . 9 ⊢ (reverse‘∅) = ∅
8	6, 7	syl6eq 2672	. . . . . . . 8 ⊢ (𝑐 = ∅ → (reverse‘𝑐) = ∅)
9	8	coeq2d 5284	. . . . . . 7 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10		co02 5649	. . . . . . 7 ⊢ (𝑀 ∘ ∅) = ∅
11	9, 10	syl6eq 2672	. . . . . 6 ⊢ (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
12	5, 11	oveq12d 6668	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
13	12	breq1d 4663	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (∅ ++ ∅) ∼ ∅))
14	13	imbi2d 330	. . 3 ⊢ (𝑐 = ∅ → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)))
15		id 22	. . . . . 6 ⊢ (𝑐 = 𝑎 → 𝑐 = 𝑎)
16		fveq2 6191	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
17	16	coeq2d 5284	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
18	15, 17	oveq12d 6668	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
19	18	breq1d 4663	. . . 4 ⊢ (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅))
20	19	imbi2d 330	. . 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 6191	. . . . . . 7 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
23	22	coeq2d 5284	. . . . . 6 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
24	21, 23	oveq12d 6668	. . . . 5 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
25	24	breq1d 4663	. . . 4 ⊢ (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
26	25	imbi2d 330	. . 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 6191	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
29	28	coeq2d 5284	. . . . . 6 ⊢ (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
30	27, 29	oveq12d 6668	. . . . 5 ⊢ (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
31	30	breq1d 4663	. . . 4 ⊢ (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
32	31	imbi2d 330	. . 3 ⊢ (𝑐 = 𝐴 → ((𝐴 ∈ 𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∼ ∅) ↔ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)))
33		wrd0 13330	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_𝑜)
34		ccatlid 13369	. . . . 5 ⊢ (∅ ∈ Word (𝐼 × 2_𝑜) → (∅ ++ ∅) = ∅)
35	33, 34	ax-mp 5	. . . 4 ⊢ (∅ ++ ∅) = ∅
36		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~_FG ‘𝐼)
37	1, 36	efger 18131	. . . . . 6 ⊢ ∼ Er 𝑊
38	37	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
39	1	efgrcl 18128	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)))
40	39	simprd 479	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2_𝑜))
41	33, 40	syl5eleqr 2708	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → ∅ ∈ 𝑊)
42	38, 41	erref 7762	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∅ ∼ ∅)
43	35, 42	syl5eqbr 4688	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∅ ++ ∅) ∼ ∅)
44	37	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ∼ Er 𝑊)
45		simprl 794	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 ∈ Word (𝐼 × 2_𝑜))
46		revcl 13510	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ Word (𝐼 × 2_𝑜) → (reverse‘𝑎) ∈ Word (𝐼 × 2_𝑜))
47	46	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (reverse‘𝑎) ∈ Word (𝐼 × 2_𝑜))
48		efgval2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
49	48	efgmf 18126	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
50		wrdco 13577	. . . . . . . . . . 11 ⊢ (((reverse‘𝑎) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_𝑜))
51	47, 49, 50	sylancl 694	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_𝑜))
52		ccatcl 13359	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_𝑜)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2_𝑜))
53	45, 51, 52	syl2anc 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2_𝑜))
54	40	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑊 = Word (𝐼 × 2_𝑜))
55	53, 54	eleqtrrd 2704	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
56		lencl 13324	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2_𝑜) → (#‘𝑎) ∈ ℕ₀)
57	56	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ ℕ₀)
58		nn0uz 11722	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
59	57, 58	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ (ℤ_≥‘0))
60		ccatlen 13360	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_𝑜)) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
61	45, 51, 60	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
62	57	nn0zd 11480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ ℤ)
63		uzid 11702	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑎) ∈ ℤ → (#‘𝑎) ∈ (ℤ_≥‘(#‘𝑎)))
64	62, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ (ℤ_≥‘(#‘𝑎)))
65		lencl 13324	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ₀)
66	51, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ₀)
67		uzaddcl 11744	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑎) ∈ (ℤ_≥‘(#‘𝑎)) ∧ (#‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ₀) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(#‘𝑎)))
68	64, 66, 67	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(#‘𝑎)))
69	61, 68	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(#‘𝑎)))
70		elfzuzb 12336	. . . . . . . . . . . 12 ⊢ ((#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((#‘𝑎) ∈ (ℤ_≥‘0) ∧ (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ_≥‘(#‘𝑎))))
71	59, 69, 70	sylanbrc 698	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
72		simprr 796	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑏 ∈ (𝐼 × 2_𝑜))
73		efgval2.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
74	1, 36, 48, 73	efgtval 18136	. . . . . . . . . . 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)”⟩⟩))
75	55, 71, 72, 74	syl3anc 1326	. . . . . . . . . 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)”⟩⟩))
76	33	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ∅ ∈ Word (𝐼 × 2_𝑜))
77	49	ffvelrni 6358	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑏) ∈ (𝐼 × 2_𝑜))
78	72, 77	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀‘𝑏) ∈ (𝐼 × 2_𝑜))
79	72, 78	s2cld 13616	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜))
80		ccatrid 13370	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ Word (𝐼 × 2_𝑜) → (𝑎 ++ ∅) = 𝑎)
81	80	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ ∅) = 𝑎)
82	81	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑎 = (𝑎 ++ ∅))
83	82	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
84		eqidd 2623	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) = (#‘𝑎))
85		hash0 13158	. . . . . . . . . . . . 13 ⊢ (#‘∅) = 0
86	85	oveq2i 6661	. . . . . . . . . . . 12 ⊢ ((#‘𝑎) + (#‘∅)) = ((#‘𝑎) + 0)
87	57	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) ∈ ℂ)
88	87	addid1d 10236	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((#‘𝑎) + 0) = (#‘𝑎))
89	86, 88	syl5req 2669	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (#‘𝑎) = ((#‘𝑎) + (#‘∅)))
90	45, 76, 51, 79, 83, 84, 89	splval2 13508	. . . . . . . . . 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))))
91	72	s1cld 13383	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_𝑜))
92		revccat 13515	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_𝑜)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
93	45, 91, 92	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
94		revs1 13514	. . . . . . . . . . . . . . . 16 ⊢ (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
95	94	oveq1i 6660	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
96	93, 95	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
97	96	coeq2d 5284	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))))
98	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜))
99		ccatco 13581	. . . . . . . . . . . . . 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))))
100	91, 47, 98, 99	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
101		s1co 13579	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (𝐼 × 2_𝑜) ∧ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
102	72, 49, 101	sylancl 694	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀‘𝑏)”⟩)
103	102	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
104	97, 100, 103	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀‘𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
105	104	oveq2d 6666	. . . . . . . . . . 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)))))
106		ccatcl 13359	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2_𝑜)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2_𝑜))
107	45, 91, 106	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2_𝑜))
108	78	s1cld 13383	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ⟨“(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_𝑜))
109		ccatass 13371	. . . . . . . . . . . 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)))))
110	107, 108, 51, 109	syl3anc 1326	. . . . . . . . . . 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)))))
111		ccatass 13371	. . . . . . . . . . . . . 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)”⟩)))
112	45, 91, 108, 111	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)))
113		df-s2 13593	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑏(𝑀‘𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩)
114	113	oveq2i 6661	. . . . . . . . . . . . 13 ⊢ (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀‘𝑏)”⟩))
115	112, 114	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀‘𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀‘𝑏)”⟩))
116	115	oveq1d 6665	. . . . . . . . . . 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))))
117	105, 110, 116	3eqtr2rd 2663	. . . . . . . . . 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”⟩)))))
118	75, 90, 117	3eqtrd 2660	. . . . . . . . 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”⟩)))))
119	1, 36, 48, 73	efgtf 18135	. . . . . . . . . . . 12 ⊢ ((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))
120	119	simprd 479	. . . . . . . . . . 11 ⊢ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_𝑜))⟶𝑊)
121		ffn 6045	. . . . . . . . . . 11 ⊢ ((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_𝑜))⟶𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2_𝑜)))
122	55, 120, 121	3syl 18	. . . . . . . . . 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)))
123		fnovrn 6809	. . . . . . . . . 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)))))
124	122, 71, 72, 123	syl3anc 1326	. . . . . . . . 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)))))
125	118, 124	eqeltrrd 2702	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
126	1, 36, 48, 73	efgi2 18138	. . . . . . . 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”⟩)))))
127	55, 125, 126	syl2anc 693	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
128	44, 127	ersym 7754	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
129	44	ertr 7757	. . . . . 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”⟩)))) ∼ ∅))
130	128, 129	mpand 711	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅))
131	130	expcom 451	. . . 4 ⊢ ((𝑎 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑏 ∈ (𝐼 × 2_𝑜)) → (𝐴 ∈ 𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∼ ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∼ ∅)))
132	131	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”⟩)))) ∼ ∅)))
133	14, 20, 26, 32, 43, 132	wrdind 13476	. 2 ⊢ (𝐴 ∈ Word (𝐼 × 2_𝑜) → (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅))
134	4, 133	mpcom 38	1 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 ⟨cop 4183 ⟨cotp 4185 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ran crn 5115 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 Er wer 7739 0cc0 9936 + caddc 9939 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 splice csplice 13296 reversecreverse 13297 ⟨“cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-efg 18122

This theorem is referenced by: efginvrel1 18141 frgpinv 18177

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