Theorem reuccats1OLD 13854

Description: Obsolete proof of reuccatpfxs1 13889 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
reuccats1.1	⊢ Ⅎ𝑣𝑋

Assertion

Ref	Expression
reuccats1OLD	⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)))

Distinct variable groups:   𝑣,𝑉,𝑥   𝑣,𝑊,𝑥   𝑥,𝑋

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem reuccats1OLD

Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2842	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉 ↔ 𝑦 ∈ Word 𝑉))
2		fveqeq2 6457	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑊) + 1) ↔ (♯‘𝑦) = ((♯‘𝑊) + 1)))
3	1, 2	anbi12d 624	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))))
4	3	cbvralv 3367	. 2 ⊢ (∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
5		reuccats1.1	. . . . 5 ⊢ Ⅎ𝑣𝑋
6	5	nfel2 2950	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋
7	5	nfel2 2950	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋
8		s1eq 13696	. . . . . 6 ⊢ (𝑣 = 𝑥 → ⟨“𝑣”⟩ = ⟨“𝑥”⟩)
9	8	oveq2d 6940	. . . . 5 ⊢ (𝑣 = 𝑥 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑥”⟩))
10	9	eleq1d 2844	. . . 4 ⊢ (𝑣 = 𝑥 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋))
11		s1eq 13696	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⟨“𝑥”⟩ = ⟨“𝑢”⟩)
12	11	oveq2d 6940	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑊 ++ ⟨“𝑥”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
13	12	eleq1d 2844	. . . 4 ⊢ (𝑥 = 𝑢 → ((𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
14	6, 7, 10, 13	reu8nf 3733	. . 3 ⊢ (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)))
15		nfv 1957	. . . . 5 ⊢ Ⅎ𝑣 𝑊 ∈ Word 𝑉
16		nfv 1957	. . . . . 6 ⊢ Ⅎ𝑣(𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
17	5, 16	nfral 3127	. . . . 5 ⊢ Ⅎ𝑣∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
18	15, 17	nfan 1946	. . . 4 ⊢ Ⅎ𝑣(𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
19		nfv 1957	. . . . 5 ⊢ Ⅎ𝑣 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)
20	5, 19	nfreu 3300	. . . 4 ⊢ Ⅎ𝑣∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)
21		simprl 761	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
22		simp-4l 773	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ Word 𝑉)
23		simpr 479	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
24	21	adantr 474	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
25		simplrr 768	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))
26		simp-4r 774	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
27		reuccats1lemOLD 13853	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑥 ∈ 𝑋 ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋) ∧ (∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢) ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))) → (𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
28	22, 23, 24, 25, 26, 27	syl32anc 1446	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
29		oveq1 6931	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 substr ⟨0, (♯‘𝑊)⟩) = ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (♯‘𝑊)⟩))
30		simpl 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
31		s1cl 13698	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ 𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
32	30, 31	anim12i 606	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
33	32	adantr 474	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
34	33	adantr 474	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
35		swrdccat1OLD 13783	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (♯‘𝑊)⟩) = 𝑊)
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (♯‘𝑊)⟩) = 𝑊)
37	29, 36	sylan9eqr 2836	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑥 substr ⟨0, (♯‘𝑊)⟩) = 𝑊)
38	37	eqcomd 2784	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩))
39	38	ex 403	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)))
40	28, 39	impbid 204	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
41	40	ralrimiva 3148	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∀𝑥 ∈ 𝑋 (𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
42		reu6i 3609	. . . . . 6 ⊢ (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩))
43	21, 41, 42	syl2anc 579	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩))
44	43	exp31 412	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (𝑣 ∈ 𝑉 → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩))))
45	18, 20, 44	rexlimd 3208	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)))
46	14, 45	syl5bi 234	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)))
47	4, 46	sylan2b 587	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (♯‘𝑊)⟩)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Ⅎwnfc 2919  ∀wral 3090  ∃wrex 3091  ∃!wreu 3092  ⟨cop 4404  ‘cfv 6137  (class class class)co 6924  0cc0 10274  1c1 10275   + caddc 10277  ♯chash 13441  Word cword 13605   ++ cconcat 13666  ⟨“cs1 13691   substr csubstr 13736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11380  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-fz 12649  df-fzo 12790  df-hash 13442  df-word 13606  df-lsw 13659  df-concat 13667  df-s1 13692  df-substr 13737

This theorem is referenced by:  reuccats1vOLD  13855  numclwlk2lem2f1oOLD  27827