Theorem hspmbl 47313

Description: Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
hspmbl.1	⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
hspmbl.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hspmbl.i	⊢ (𝜑 → 𝐾 ∈ 𝑋)
hspmbl.y	⊢ (𝜑 → 𝑌 ∈ ℝ)

Assertion

Ref	Expression
hspmbl	⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))

Distinct variable groups:   𝐾,𝑙,𝑥,𝑦   𝑋,𝑙,𝑥,𝑦   𝑌,𝑙,𝑥,𝑦   𝜑,𝑙   𝑘,𝑙,𝑥,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑘)   𝐻(𝑥,𝑦,𝑘,𝑙)   𝐾(𝑘)   𝑋(𝑘)   𝑌(𝑘)

Proof of Theorem hspmbl

Dummy variables 𝑎 𝑗 𝑝 𝑡 𝑏 ℎ 𝑐 𝑟 𝑠 𝑖 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hspmbl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
2	1	ovnome 47257	. . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3		eqid 2761	. . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋)
4		eqid 2761	. . 3 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5		ovex 7443	. . . . . . . . 9 ⊢ (-∞(,)𝑌) ∈ V
6		reex 11190	. . . . . . . . 9 ⊢ ℝ ∈ V
7	5, 6	ifex 4537	. . . . . . . 8 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
8	7	ixpssmap 8929	. . . . . . 7 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋)
9		iftrue 4492	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10		ioossre 13433	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑌) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
12	9, 11	eqsstrd 3970	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13		iffalse 4495	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14		ssid 3958	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → ℝ ⊆ ℝ)
16	13, 15	eqsstrd 3970	. . . . . . . . . . 11 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
17	12, 16	pm2.61i 184	. . . . . . . . . 10 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
18	17	rgenw 3081	. . . . . . . . 9 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19		iunss 5008	. . . . . . . . 9 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
20	18, 19	mpbir 234	. . . . . . . 8 ⊢ ∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21		mapss 8886	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ ∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋))
22	6, 20, 21	mp2an 704	. . . . . . 7 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)
23	8, 22	sstri 3945	. . . . . 6 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)
24	7	rgenw 3081	. . . . . . . 8 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25		ixpexg 8919	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V → X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V)
26	24, 25	ax-mp 5	. . . . . . 7 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
27		elpwg 4564	. . . . . . 7 ⊢ (X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V → (X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋) ↔ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)))
28	26, 27	ax-mp 5	. . . . . 6 ⊢ (X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋) ↔ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋))
29	23, 28	mpbir 234	. . . . 5 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋))
31		hspmbl.1	. . . . . . 7 ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
32		equid 2040	. . . . . . . . 9 ⊢ 𝑥 = 𝑥
33		eqid 2761	. . . . . . . . 9 ⊢ ℝ = ℝ
34		equequ1 2053	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑝 → (𝑘 = 𝑙 ↔ 𝑝 = 𝑙))
35	34	ifbid 4510	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
36	35	cbvixpv 8912	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
37	32, 33, 36	mpoeq123i 7486	. . . . . . . 8 ⊢ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
38	37	mpteq2i 5206	. . . . . . 7 ⊢ (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ))) = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)))
39	31, 38	eqtri 2784	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)))
40		hspmbl.i	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
41		hspmbl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
42	39, 1, 40, 41	hspval 47293	. . . . 5 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) = X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
43	1	ovnf 47247	. . . . . . . . 9 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞))
44	43	fdmd 6716	. . . . . . . 8 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
45	44	unieqd 4884	. . . . . . 7 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑m 𝑋))
46		unipw 5431	. . . . . . . 8 ⊢ ∪ 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋)
47	46	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∪ 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋))
48	45, 47	eqtrd 2796	. . . . . 6 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋))
49	48	pweqd 4578	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
50	42, 49	eleq12d 2855	. . . 4 ⊢ (𝜑 → ((𝐾(𝐻‘𝑋)𝑌) ∈ 𝒫 ∪ dom (voln*‘𝑋) ↔ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋)))
51	30, 50	mpbird 260	. . 3 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ 𝒫 ∪ dom (voln*‘𝑋))
52		simpl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝜑)
53		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋))
54	52, 49	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
55	53, 54	eleqtrd 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋))
56	1	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑋 ∈ Fin)
57		inss1 4188	. . . . . . . . . . . . 13 ⊢ (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝑎
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝑎)
59		elpwi 4568	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → 𝑎 ⊆ (ℝ ↑m 𝑋))
60	58, 59	sstrd 3946	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
61	60	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
62	56, 61	ovnxrcl 47253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
63	59	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑎 ⊆ (ℝ ↑m 𝑋))
64	63	ssdifssd 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
65	56, 64	ovnxrcl 47253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
66	62, 65	xaddcld 13326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ*)
67		pnfge 13154	. . . . . . . 8 ⊢ ((((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ* → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ +∞)
68	66, 67	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ +∞)
69	68	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ +∞)
70		id 23	. . . . . . . 8 ⊢ (((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) = +∞)
71	70	eqcomd 2767	. . . . . . 7 ⊢ (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
72	71	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
73	69, 72	breqtrd 5136	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
74		simpl 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)))
75	56, 63	ovncl 47251	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
76	75	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
77		neqne 2964	. . . . . . . 8 ⊢ (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
78	77	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
79		ge0xrre 46217	. . . . . . 7 ⊢ ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
80	76, 78, 79	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
81	56	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
82	40	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾 ∈ 𝑋)
83	41	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
84		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
85	63	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑m 𝑋))
86		sseq1 3961	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
87	86	rabbidv 3421	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})
88	87	cbvmptv 5214	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)}) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})
89		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → 𝑖 = ℎ)
90	89	coeq2d 5848	. . . . . . . . . . 11 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ℎ))
91	90	fveq1d 6883	. . . . . . . . . 10 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ ℎ)‘𝑝))
92	91	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ ℎ)‘𝑝)))
93	92	prodeq2dv 15976	. . . . . . . 8 ⊢ (𝑖 = ℎ → ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
94	93	cbvmptv 5214	. . . . . . 7 ⊢ (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
95		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑝 → (([,) ∘ (𝑚‘𝑖))‘𝑛) = (([,) ∘ (𝑚‘𝑖))‘𝑝))
96	95	cbvixpv 8912	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝)
97	96	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ℎ → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝))
98		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = ℎ → (𝑚‘𝑖) = (ℎ‘𝑖))
99	98	coeq2d 5848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ℎ → ([,) ∘ (𝑚‘𝑖)) = ([,) ∘ (ℎ‘𝑖)))
100	99	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ℎ → (([,) ∘ (𝑚‘𝑖))‘𝑝) = (([,) ∘ (ℎ‘𝑖))‘𝑝))
101	100	ixpeq2dv 8910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ℎ → X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
102	97, 101	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = ℎ → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
103	102	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = ℎ ∧ 𝑖 ∈ ℕ) → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
104	103	iuneq2dv 4980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = ℎ → ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
105	104	sseq2d 3968	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = ℎ → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) ↔ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)))
106	105	cbvrabv 3424	. . . . . . . . . . . . . . . . 17 ⊢ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)} = {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)}
107		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑙 → (ℎ‘𝑖) = (𝑙‘𝑖))
108	107	coeq2d 5848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑙 → ([,) ∘ (ℎ‘𝑖)) = ([,) ∘ (𝑙‘𝑖)))
109	108	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑙 → (([,) ∘ (ℎ‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑖))‘𝑝))
110	109	ixpeq2dv 8910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑙 → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
111	110	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = 𝑙 ∧ 𝑖 ∈ ℕ) → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
112	111	iuneq2dv 4980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
113		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (𝑙‘𝑖) = (𝑙‘𝑗))
114	113	coeq2d 5848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → ([,) ∘ (𝑙‘𝑖)) = ([,) ∘ (𝑙‘𝑗)))
115	114	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑗 → (([,) ∘ (𝑙‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑗))‘𝑝))
116	115	ixpeq2dv 8910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
117	116	cbviunv 5002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)
118	117	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
119	112, 118	eqtrd 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
120	119	sseq2d 3968	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑙 → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
121	120	cbvrabv 3424	. . . . . . . . . . . . . . . . 17 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)}
122	106, 121	eqtri 2784	. . . . . . . . . . . . . . . 16 ⊢ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)}
123	122	mpteq2i 5206	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)}) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})
124	123	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)}) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)}))
125		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → 𝑐 = 𝑏)
126	124, 125	fveq12d 6888	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) = ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏))
127	126	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ↔ 𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏)))
128		2fveq3 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
129	128	cbvprodv 15968	. . . . . . . . . . . . . . . . . . 19 ⊢ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
130	129	mpteq2i 5206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))
131	130	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))))
132		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑡‘𝑚) = (𝑡‘𝑗))
133	131, 132	fveq12d 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))
134	133	cbvmptv 5214	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚))) = (𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))
135	134	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚))) = (𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗))))
136	135	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))))
137		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
138	137	oveq1d 7425	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
139	136, 138	breq12d 5121	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → ((Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)))
140	127, 139	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → ((𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ∧ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)) ↔ (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))))
141	140	rabbidva2 3416	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)} = {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)})
142	141	mpteq2dv 5204	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)}) = (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}))
143		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑟 → ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) = ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏))
144	143	eleq2d 2847	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ↔ 𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏)))
145		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
146	145	breq2d 5120	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑟 → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)))
147	144, 146	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑟 → ((𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)) ↔ (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))))
148	147	rabbidva2 3416	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑟 → {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)} = {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)})
149	148	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)})
150	149	a1i 11	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
151	142, 150	eqtrd 2796	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
152	151	cbvmptv 5214	. . . . . . 7 ⊢ (𝑐 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡‘𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)})) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
153		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑚 = 𝑝 → (1st ‘((𝑡‘𝑗)‘𝑚)) = (1st ‘((𝑡‘𝑗)‘𝑝)))
154	153	cbvmptv 5214	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝)))
155	154	mpteq2i 5206	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝))))
156		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑡‘𝑖) = (𝑡‘𝑗))
157	156	fveq1d 6883	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑡‘𝑖)‘𝑚) = ((𝑡‘𝑗)‘𝑚))
158	157	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (2nd ‘((𝑡‘𝑖)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑚)))
159	158	mpteq2dv 5204	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑚))))
160		2fveq3 6886	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑝 → (2nd ‘((𝑡‘𝑗)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑝)))
161	160	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝)))
162	161	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝))))
163	159, 162	eqtrd 2796	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝))))
164	163	cbvmptv 5214	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝))))
165	39, 81, 82, 83, 84, 85, 88, 94, 152, 155, 164	hspmbllem3 47312	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
166	74, 80, 165	syl2anc 595	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
167	73, 166	pm2.61dan 824	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
168	52, 55, 167	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
169	2, 3, 4, 51, 168	caragenel2d 47216	. 2 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
170	1	dmvon 47290	. . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
171	170	eqcomd 2767	. 2 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
172	169, 171	eleqtrd 2863	1 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  {crab 3414  Vcvv 3453   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ifcif 4486  𝒫 cpw 4561  ∪ cuni 4871  ∪ ciun 4955   class class class wbr 5108   ↦ cmpt 5191   × cxp 5659  dom cdm 5661   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7983  2^nd c2nd 7984   ↑_m cmap 8823  Xcixp 8894  Fincfn 8942  ℝcr 11098  0cc0 11099  +∞cpnf 11239  -∞cmnf 11240  ℝ^*cxr 11241   ≤ cle 11243  ℕcn 12232  ℝ⁺crp 13015   +_𝑒 cxad 13134  (,)cioo 13371  [,)cico 13373  [,]cicc 13374  ∏cprod 15957  volcvol 25601  Σ^{^}csumge0 47046  CaraGenccaragen 47175  voln*covoln 47220  volncvoln 47222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cc 10418  ax-ac2 10446  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-acn 9927  df-ac 10099  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13683  df-fl 13825  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-rlim 15540  df-sum 15738  df-prod 15958  df-rest 17474  df-topgen 17495  df-psmet 21493  df-xmet 21494  df-met 21495  df-bl 21496  df-mopn 21497  df-top 23030  df-topon 23047  df-bases 23082  df-cmp 23523  df-ovol 25602  df-vol 25603  df-sumge0 47047  df-ome 47174  df-caragen 47176  df-ovoln 47221  df-voln 47223

This theorem is referenced by:  hoimbllem  47314