Theorem ennnfonelemp1 12566

Description: Lemma for ennnfone 12585. Value of 𝐻 at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemp1.p	⊢ (𝜑 → 𝑃 ∈ ℕ0)

Assertion

Ref	Expression
ennnfonelemp1	⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))

Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑥,𝐹,𝑦   𝑗,𝐺   𝑥,𝐻,𝑦   𝑗,𝐽   𝑥,𝑁,𝑦   𝑃,𝑗,𝑥,𝑦   𝜑,𝑗,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐴(𝑘,𝑛)   𝑃(𝑘,𝑛)   𝐹(𝑗,𝑘,𝑛)   𝐺(𝑥,𝑦,𝑘,𝑛)   𝐻(𝑗,𝑘,𝑛)   𝐽(𝑥,𝑦,𝑘,𝑛)   𝑁(𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelemp1

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemp1.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
2		nn0uz 9630	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2286	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘0))
4		ennnfonelemh.dceq	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
5		ennnfonelemh.f	. . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
6		ennnfonelemh.ne	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
7		ennnfonelemh.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
8		ennnfonelemh.n	. . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
9		ennnfonelemh.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
10		ennnfonelemh.h	. . . . 5 ⊢ 𝐻 = seq0(𝐺, 𝐽)
11	4, 5, 6, 7, 8, 9, 10	ennnfonelemj0 12561	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
12	4, 5, 6, 7, 8, 9, 10	ennnfonelemg 12563	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
13	4, 5, 6, 7, 8, 9, 10	ennnfonelemjn 12562	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
14	3, 11, 12, 13	seqp1cd 10544	. . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘(𝑃 + 1)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
15	10	fveq1i 5556	. . . 4 ⊢ (𝐻‘(𝑃 + 1)) = (seq0(𝐺, 𝐽)‘(𝑃 + 1))
16	15	a1i 9	. . 3 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = (seq0(𝐺, 𝐽)‘(𝑃 + 1)))
17	10	fveq1i 5556	. . . . 5 ⊢ (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃)
18	17	a1i 9	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃))
19		eqeq1 2200	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (𝑥 = 0 ↔ (𝑃 + 1) = 0))
20		fvoveq1 5942	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘((𝑃 + 1) − 1)))
21	19, 20	ifbieq2d 3582	. . . . . 6 ⊢ (𝑥 = (𝑃 + 1) → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
22		peano2nn0 9283	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ∈ ℕ0)
23	1, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑃 + 1) ∈ ℕ0)
24		nn0p1gt0 9272	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0 → 0 < (𝑃 + 1))
25	24	gt0ne0d 8533	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ≠ 0)
26	25	neneqd 2385	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ¬ (𝑃 + 1) = 0)
27	26	iffalsed 3568	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘((𝑃 + 1) − 1)))
28		nn0cn 9253	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℂ)
29		1cnd 8037	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 1 ∈ ℂ)
30	28, 29	pncand 8333	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ((𝑃 + 1) − 1) = 𝑃)
31	30	fveq2d 5559	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘((𝑃 + 1) − 1)) = (◡𝑁‘𝑃))
32	27, 31	eqtrd 2226	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
33	8	frechashgf1o 10502	. . . . . . . . . . 11 ⊢ 𝑁:ω–1-1-onto→ℕ0
34		f1ocnv 5514	. . . . . . . . . . 11 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
35	33, 34	ax-mp 5	. . . . . . . . . 10 ⊢ ◡𝑁:ℕ0–1-1-onto→ω
36		f1of 5501	. . . . . . . . . 10 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0⟶ω)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → ◡𝑁:ℕ0⟶ω)
38		id 19	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℕ0)
39	37, 38	ffvelcdmd 5695	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘𝑃) ∈ ω)
40	32, 39	eqeltrd 2270	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) ∈ ω)
41	1, 40	syl 14	. . . . . 6 ⊢ (𝜑 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) ∈ ω)
42	9, 21, 23, 41	fvmptd3 5652	. . . . 5 ⊢ (𝜑 → (𝐽‘(𝑃 + 1)) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
43	1, 32	syl 14	. . . . 5 ⊢ (𝜑 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
44	42, 43	eqtr2d 2227	. . . 4 ⊢ (𝜑 → (◡𝑁‘𝑃) = (𝐽‘(𝑃 + 1)))
45	18, 44	oveq12d 5937	. . 3 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
46	14, 16, 45	3eqtr4d 2236	. 2 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)))
47	4, 5, 6, 7, 8, 9, 10	ennnfonelemh 12564	. . . 4 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
48	47, 1	ffvelcdmd 5695	. . 3 ⊢ (𝜑 → (𝐻‘𝑃) ∈ (𝐴 ↑pm ω))
49	1, 39	syl 14	. . 3 ⊢ (𝜑 → (◡𝑁‘𝑃) ∈ ω)
50	48	elexd 2773	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) ∈ V)
51		dmexg 4927	. . . . . . . 8 ⊢ ((𝐻‘𝑃) ∈ V → dom (𝐻‘𝑃) ∈ V)
52	50, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → dom (𝐻‘𝑃) ∈ V)
53		fof 5477	. . . . . . . . 9 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
54	5, 53	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω⟶𝐴)
55	54, 49	ffvelcdmd 5695	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴)
56		opexg 4258	. . . . . . 7 ⊢ ((dom (𝐻‘𝑃) ∈ V ∧ (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴) → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
57	52, 55, 56	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
58		snexg 4214	. . . . . 6 ⊢ (⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V → {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V)
59	57, 58	syl 14	. . . . 5 ⊢ (𝜑 → {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V)
60		unexg 4475	. . . . 5 ⊢ (((𝐻‘𝑃) ∈ V ∧ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}) ∈ V)
61	50, 59, 60	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}) ∈ V)
62	4, 5, 49	ennnfonelemdc 12559	. . . 4 ⊢ (𝜑 → DECID (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)))
63	50, 61, 62	ifcldcd 3594	. . 3 ⊢ (𝜑 → if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})) ∈ V)
64		id 19	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → 𝑥 = (𝐻‘𝑃))
65		dmeq 4863	. . . . . . . 8 ⊢ (𝑥 = (𝐻‘𝑃) → dom 𝑥 = dom (𝐻‘𝑃))
66	65	opeq1d 3811	. . . . . . 7 ⊢ (𝑥 = (𝐻‘𝑃) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩)
67	66	sneqd 3632	. . . . . 6 ⊢ (𝑥 = (𝐻‘𝑃) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})
68	64, 67	uneq12d 3315	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}))
69	64, 68	ifeq12d 3577	. . . 4 ⊢ (𝑥 = (𝐻‘𝑃) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})) = if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})))
70		fveq2 5555	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹‘𝑦) = (𝐹‘(◡𝑁‘𝑃)))
71		imaeq2 5002	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝑁‘𝑃)))
72	70, 71	eleq12d 2264	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐹‘𝑦) ∈ (𝐹 “ 𝑦) ↔ (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃))))
73	70	opeq2d 3812	. . . . . . 7 ⊢ (𝑦 = (◡𝑁‘𝑃) → ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩)
74	73	sneqd 3632	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})
75	74	uneq2d 3314	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}))
76	72, 75	ifbieq2d 3582	. . . 4 ⊢ (𝑦 = (◡𝑁‘𝑃) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
77	69, 76, 7	ovmpog 6054	. . 3 ⊢ (((𝐻‘𝑃) ∈ (𝐴 ↑pm ω) ∧ (◡𝑁‘𝑃) ∈ ω ∧ if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})) ∈ V) → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
78	48, 49, 63, 77	syl3anc 1249	. 2 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
79	46, 78	eqtrd 2226	1 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))

Syntax hints:   → wi 4  DECID wdc 835   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472  ∃wrex 2473  {crab 2476  Vcvv 2760   ∪ cun 3152  ∅c0 3447  ifcif 3558  {csn 3619  ⟨cop 3622   ↦ cmpt 4091  suc csuc 4397  ωcom 4623  ^◡ccnv 4659  dom cdm 4660   “ cima 4663  ⟶wf 5251  –onto→wfo 5253  –1-1-onto→wf1o 5254  ‘cfv 5255  (class class class)co 5919   ∈ cmpo 5921  freccfrec 6445   ↑_pm cpm 6705  0cc0 7874  1c1 7875   + caddc 7877   − cmin 8192  ℕ₀cn0 9243  ℤcz 9320  ℤ_≥cuz 9595  seqcseq 10521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pm 6707  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522

This theorem is referenced by:  ennnfonelem1  12567  ennnfonelemhdmp1  12569  ennnfonelemss  12570  ennnfonelemkh  12572  ennnfonelemhf1o  12573