Theorem depindlem3 16920

Description: Lemma for depind 16921. (Contributed by Matthew House, 14-Apr-2026.)

Hypotheses

Ref	Expression
depind.p	⊢ (𝜑 → 𝑃:ℕ0⟶V)
depind.0	⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
depind.h	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
depindlem1.4	⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))

Assertion

Ref	Expression
depindlem3	⊢ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

Distinct variable groups:   𝑓,ℎ,𝑛,𝑥   𝜑,𝑓   𝐴,𝑓,𝑚,𝑛   𝑛,𝐹   𝑓,𝐻,𝑚,𝑛   𝑃,𝑓,𝑛

Allowed substitution hints:   𝜑(𝑥, ℎ, 𝑚, 𝑛)   𝐴(𝑥, ℎ)   𝑃(𝑥, ℎ, 𝑚)   𝐹(𝑥, 𝑓, ℎ, 𝑚)   𝐻(𝑥, ℎ)

Proof of Theorem depindlem3

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

Step	Hyp	Ref	Expression
1		ixpfn 6986	. . . . 5 ⊢ (𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛) → 𝑓 Fn ℕ0)
2	1	ad2antlr 493	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝑓 Fn ℕ0)
3		depind.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃:ℕ0⟶V)
4		depind.0	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
5		depind.h	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
6		depindlem1.4	. . . . . . . 8 ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))
7	3, 4, 5, 6	depindlem1 16918	. . . . . . 7 ⊢ (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
8	7	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
9	8	simp1d 1040	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝐹:ℕ0⟶V)
10	9	ffnd 5534	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝐹 Fn ℕ0)
11		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑓‘𝑦) = (𝑓‘0))
12		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
13	11, 12	eqeq12d 2253	. . . . . . 7 ⊢ (𝑦 = 0 → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘0) = (𝐹‘0)))
14	13	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 0 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = (𝐹‘0))))
15		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝑓‘𝑦) = (𝑓‘𝑘))
16		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
17	15, 16	eqeq12d 2253	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘𝑘) = (𝐹‘𝑘)))
18	17	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 𝑘 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘))))
19		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝑓‘𝑦) = (𝑓‘(𝑘 + 1)))
20		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝐹‘𝑦) = (𝐹‘(𝑘 + 1)))
21	19, 20	eqeq12d 2253	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1))))
22	21	imbi2d 230	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
23		simprl 535	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = 𝐴)
24	8	simp2d 1041	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝐹‘0) = 𝐴)
25	23, 24	eqtr4d 2274	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = (𝐹‘0))
26		fveq2 5695	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑘) = (𝐹‘𝑘) → ((𝐻‘𝑘)‘(𝑓‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
27	26	ad2antlr 493	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → ((𝐻‘𝑘)‘(𝑓‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
28		simplrr 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) → ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))
29		fvoveq1 6108	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘(𝑛 + 1)) = (𝑓‘(𝑘 + 1)))
30		fveq2 5695	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐻‘𝑛) = (𝐻‘𝑘))
31		fveq2 5695	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
32	30, 31	fveq12d 5702	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝑓‘𝑛)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
33	29, 32	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)) ↔ (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘))))
34	33	rspccva 2928	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
35	28, 34	sylan 283	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
36	8	simp3d 1042	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
37	36	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
38		fvoveq1 6108	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
39		fveq2 5695	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
40	30, 39	fveq12d 5702	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝐹‘𝑛)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
41	38, 40	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ↔ (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘))))
42	41	rspccva 2928	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
43	37, 42	sylan 283	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
44	27, 35, 43	3eqtr4d 2281	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))
45	44	exp31 364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ((𝑓‘𝑘) = (𝐹‘𝑘) → (𝑘 ∈ ℕ0 → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
46	45	com3r 79	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ((𝑓‘𝑘) = (𝐹‘𝑘) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
47	46	a2d 26	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘)) → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
48	14, 18, 22, 18, 25, 47	nn0ind 9765	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘)))
49	48	impcom 125	. . . 4 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ 𝑘 ∈ ℕ0) → (𝑓‘𝑘) = (𝐹‘𝑘))
50	2, 10, 49	eqfnfvd 5809	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝑓 = 𝐹)
51	50	ex 115	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) → (((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))
52	51	ralrimiva 2623	1 ⊢ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  ifcif 3638   ↦ cmpt 4192   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  Xcixp 6980  0cc0 8180  1c1 8181   + caddc 8183   − cmin 8499  ℕ₀cn0 9568  seqcseq 10899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-ixp 6981  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-seqfrec 10900

This theorem is used by:  depind  16921