Theorem ennnfonelemp1 12276

Description: Lemma for ennnfone 12295. 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 9491	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2257	. . . 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 12271	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
12	4, 5, 6, 7, 8, 9, 10	ennnfonelemg 12273	. . . 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 12272	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
14	3, 11, 12, 13	seqp1cd 10391	. . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘(𝑃 + 1)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
15	10	fveq1i 5481	. . . 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 5481	. . . . 5 ⊢ (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃)
18	17	a1i 9	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃))
19		eqeq1 2171	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (𝑥 = 0 ↔ (𝑃 + 1) = 0))
20		fvoveq1 5859	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘((𝑃 + 1) − 1)))
21	19, 20	ifbieq2d 3539	. . . . . 6 ⊢ (𝑥 = (𝑃 + 1) → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
22		peano2nn0 9145	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ∈ ℕ0)
23	1, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑃 + 1) ∈ ℕ0)
24		nn0p1gt0 9134	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0 → 0 < (𝑃 + 1))
25	24	gt0ne0d 8401	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ≠ 0)
26	25	neneqd 2355	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ¬ (𝑃 + 1) = 0)
27	26	iffalsed 3525	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘((𝑃 + 1) − 1)))
28		nn0cn 9115	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℂ)
29		1cnd 7906	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 1 ∈ ℂ)
30	28, 29	pncand 8201	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ((𝑃 + 1) − 1) = 𝑃)
31	30	fveq2d 5484	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘((𝑃 + 1) − 1)) = (◡𝑁‘𝑃))
32	27, 31	eqtrd 2197	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
33	8	frechashgf1o 10353	. . . . . . . . . . 11 ⊢ 𝑁:ω–1-1-onto→ℕ0
34		f1ocnv 5439	. . . . . . . . . . 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 5426	. . . . . . . . . 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	ffvelrnd 5615	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘𝑃) ∈ ω)
40	32, 39	eqeltrd 2241	. . . . . . 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 5573	. . . . 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 2198	. . . 4 ⊢ (𝜑 → (◡𝑁‘𝑃) = (𝐽‘(𝑃 + 1)))
45	18, 44	oveq12d 5854	. . 3 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
46	14, 16, 45	3eqtr4d 2207	. 2 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)))
47	4, 5, 6, 7, 8, 9, 10	ennnfonelemh 12274	. . . 4 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
48	47, 1	ffvelrnd 5615	. . 3 ⊢ (𝜑 → (𝐻‘𝑃) ∈ (𝐴 ↑pm ω))
49	1, 39	syl 14	. . 3 ⊢ (𝜑 → (◡𝑁‘𝑃) ∈ ω)
50	48	elexd 2734	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) ∈ V)
51		dmexg 4862	. . . . . . . 8 ⊢ ((𝐻‘𝑃) ∈ V → dom (𝐻‘𝑃) ∈ V)
52	50, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → dom (𝐻‘𝑃) ∈ V)
53		fof 5404	. . . . . . . . 9 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
54	5, 53	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω⟶𝐴)
55	54, 49	ffvelrnd 5615	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴)
56		opexg 4200	. . . . . . 7 ⊢ ((dom (𝐻‘𝑃) ∈ V ∧ (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴) → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
57	52, 55, 56	syl2anc 409	. . . . . 6 ⊢ (𝜑 → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
58		snexg 4157	. . . . . 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 4415	. . . . 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 409	. . . 4 ⊢ (𝜑 → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}) ∈ V)
62	4, 5, 49	ennnfonelemdc 12269	. . . 4 ⊢ (𝜑 → DECID (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)))
63	50, 61, 62	ifcldcd 3550	. . 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 4798	. . . . . . . 8 ⊢ (𝑥 = (𝐻‘𝑃) → dom 𝑥 = dom (𝐻‘𝑃))
66	65	opeq1d 3758	. . . . . . 7 ⊢ (𝑥 = (𝐻‘𝑃) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩)
67	66	sneqd 3583	. . . . . 6 ⊢ (𝑥 = (𝐻‘𝑃) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})
68	64, 67	uneq12d 3272	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}))
69	64, 68	ifeq12d 3534	. . . 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 5480	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹‘𝑦) = (𝐹‘(◡𝑁‘𝑃)))
71		imaeq2 4936	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝑁‘𝑃)))
72	70, 71	eleq12d 2235	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐹‘𝑦) ∈ (𝐹 “ 𝑦) ↔ (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃))))
73	70	opeq2d 3759	. . . . . . 7 ⊢ (𝑦 = (◡𝑁‘𝑃) → ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩)
74	73	sneqd 3583	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})
75	74	uneq2d 3271	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}))
76	72, 75	ifbieq2d 3539	. . . 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 5967	. . 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 1227	. 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 2197	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 824   = wceq 1342   ∈ wcel 2135   ≠ wne 2334  ∀wral 2442  ∃wrex 2443  {crab 2446  Vcvv 2721   ∪ cun 3109  ∅c0 3404  ifcif 3515  {csn 3570  ⟨cop 3573   ↦ cmpt 4037  suc csuc 4337  ωcom 4561  ^◡ccnv 4597  dom cdm 4598   “ cima 4601  ⟶wf 5178  –onto→wfo 5180  –1-1-onto→wf1o 5181  ‘cfv 5182  (class class class)co 5836   ∈ cmpo 5838  freccfrec 6349   ↑_pm cpm 6606  0cc0 7744  1c1 7745   + caddc 7747   − cmin 8060  ℕ₀cn0 9105  ℤcz 9182  ℤ_≥cuz 9457  seqcseq 10370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pm 6608  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-seqfrec 10371

This theorem is referenced by:  ennnfonelem1  12277  ennnfonelemhdmp1  12279  ennnfonelemss  12280  ennnfonelemkh  12282  ennnfonelemhf1o  12283