Theorem hspmbl 46550

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 46494	. . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3		eqid 2740	. . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋)
4		eqid 2740	. . 3 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5		ovex 7481	. . . . . . . . 9 ⊢ (-∞(,)𝑌) ∈ V
6		reex 11275	. . . . . . . . 9 ⊢ ℝ ∈ V
7	5, 6	ifex 4598	. . . . . . . 8 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
8	7	ixpssmap 8990	. . . . . . 7 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋)
9		iftrue 4554	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10		ioossre 13468	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑌) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
12	9, 11	eqsstrd 4047	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13		iffalse 4557	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14		ssid 4031	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → ℝ ⊆ ℝ)
16	13, 15	eqsstrd 4047	. . . . . . . . . . 11 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
17	12, 16	pm2.61i 182	. . . . . . . . . 10 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
18	17	rgenw 3071	. . . . . . . . 9 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19		iunss 5068	. . . . . . . . 9 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
20	18, 19	mpbir 231	. . . . . . . 8 ⊢ ∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21		mapss 8947	. . . . . . . 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 691	. . . . . . 7 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)
23	8, 22	sstri 4018	. . . . . 6 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)
24	7	rgenw 3071	. . . . . . . 8 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25		ixpexg 8980	. . . . . . . 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 4625	. . . . . . 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 231	. . . . 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 2011	. . . . . . . . 9 ⊢ 𝑥 = 𝑥
33		eqid 2740	. . . . . . . . 9 ⊢ ℝ = ℝ
34		equequ1 2024	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑝 → (𝑘 = 𝑙 ↔ 𝑝 = 𝑙))
35	34	ifbid 4571	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
36	35	cbvixpv 8973	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
37	32, 33, 36	mpoeq123i 7526	. . . . . . . 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 5271	. . . . . . 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 2768	. . . . . 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 46530	. . . . 5 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) = X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
43	1	ovnf 46484	. . . . . . . . 9 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞))
44	43	fdmd 6757	. . . . . . . 8 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
45	44	unieqd 4944	. . . . . . 7 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑m 𝑋))
46		unipw 5470	. . . . . . . 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 2780	. . . . . 6 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋))
49	48	pweqd 4639	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
50	42, 49	eleq12d 2838	. . . 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 257	. . 3 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ 𝒫 ∪ dom (voln*‘𝑋))
52		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝜑)
53		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋))
54	52, 49	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
55	53, 54	eleqtrd 2846	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋))
56	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑋 ∈ Fin)
57		inss1 4258	. . . . . . . . . . . . 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 4629	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → 𝑎 ⊆ (ℝ ↑m 𝑋))
60	58, 59	sstrd 4019	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
61	60	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
62	56, 61	ovnxrcl 46490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
63	59	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑎 ⊆ (ℝ ↑m 𝑋))
64	63	ssdifssd 4170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
65	56, 64	ovnxrcl 46490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
66	62, 65	xaddcld 13363	. . . . . . . 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 13193	. . . . . . . 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 17	. . . . . . 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 480	. . . . . 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 22	. . . . . . . 8 ⊢ (((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) = +∞)
71	70	eqcomd 2746	. . . . . . 7 ⊢ (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
72	71	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
73	69, 72	breqtrd 5192	. . . . 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 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)))
75	56, 63	ovncl 46488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
76	75	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
77		neqne 2954	. . . . . . . 8 ⊢ (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
78	77	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
79		ge0xrre 45449	. . . . . . 7 ⊢ ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
80	76, 78, 79	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
81	56	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
82	40	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾 ∈ 𝑋)
83	41	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
84		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
85	63	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑m 𝑋))
86		sseq1 4034	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
87	86	rabbidv 3451	. . . . . . . 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 5279	. . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → 𝑖 = ℎ)
90	89	coeq2d 5887	. . . . . . . . . . 11 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ℎ))
91	90	fveq1d 6922	. . . . . . . . . 10 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ ℎ)‘𝑝))
92	91	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ ℎ)‘𝑝)))
93	92	prodeq2dv 15970	. . . . . . . 8 ⊢ (𝑖 = ℎ → ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
94	93	cbvmptv 5279	. . . . . . 7 ⊢ (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
95		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑝 → (([,) ∘ (𝑚‘𝑖))‘𝑛) = (([,) ∘ (𝑚‘𝑖))‘𝑝))
96	95	cbvixpv 8973	. . . . . . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = ℎ → (𝑚‘𝑖) = (ℎ‘𝑖))
99	98	coeq2d 5887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ℎ → ([,) ∘ (𝑚‘𝑖)) = ([,) ∘ (ℎ‘𝑖)))
100	99	fveq1d 6922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ℎ → (([,) ∘ (𝑚‘𝑖))‘𝑝) = (([,) ∘ (ℎ‘𝑖))‘𝑝))
101	100	ixpeq2dv 8971	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ℎ → X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
102	97, 101	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = ℎ → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = ℎ ∧ 𝑖 ∈ ℕ) → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
104	103	iuneq2dv 5039	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = ℎ → ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
105	104	sseq2d 4041	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = ℎ → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) ↔ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)))
106	105	cbvrabv 3454	. . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑙 → (ℎ‘𝑖) = (𝑙‘𝑖))
108	107	coeq2d 5887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑙 → ([,) ∘ (ℎ‘𝑖)) = ([,) ∘ (𝑙‘𝑖)))
109	108	fveq1d 6922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑙 → (([,) ∘ (ℎ‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑖))‘𝑝))
110	109	ixpeq2dv 8971	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑙 → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
111	110	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = 𝑙 ∧ 𝑖 ∈ ℕ) → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
112	111	iuneq2dv 5039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
113		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (𝑙‘𝑖) = (𝑙‘𝑗))
114	113	coeq2d 5887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → ([,) ∘ (𝑙‘𝑖)) = ([,) ∘ (𝑙‘𝑗)))
115	114	fveq1d 6922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑗 → (([,) ∘ (𝑙‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑗))‘𝑝))
116	115	ixpeq2dv 8971	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
117	116	cbviunv 5063	. . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
120	119	sseq2d 4041	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑙 → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
121	120	cbvrabv 3454	. . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . 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 5271	. . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → 𝑐 = 𝑏)
126	124, 125	fveq12d 6927	. . . . . . . . . . . . 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 2830	. . . . . . . . . . . 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 6925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
129	128	cbvprodv 15962	. . . . . . . . . . . . . . . . . . 19 ⊢ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
130	129	mpteq2i 5271	. . . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑡‘𝑚) = (𝑡‘𝑗))
133	131, 132	fveq12d 6927	. . . . . . . . . . . . . . . 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 5279	. . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
138	137	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
139	136, 138	breq12d 5179	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 3445	. . . . . . . . . 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 5268	. . . . . . . . 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 2741	. . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
146	145	breq2d 5178	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3445	. . . . . . . . . . 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 5279	. . . . . . . . . 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 2780	. . . . . . . 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 5279	. . . . . . 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 6925	. . . . . . . . 9 ⊢ (𝑚 = 𝑝 → (1st ‘((𝑡‘𝑗)‘𝑚)) = (1st ‘((𝑡‘𝑗)‘𝑝)))
154	153	cbvmptv 5279	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝)))
155	154	mpteq2i 5271	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝))))
156		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑡‘𝑖) = (𝑡‘𝑗))
157	156	fveq1d 6922	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑡‘𝑖)‘𝑚) = ((𝑡‘𝑗)‘𝑚))
158	157	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (2nd ‘((𝑡‘𝑖)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑚)))
159	158	mpteq2dv 5268	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑚))))
160		2fveq3 6925	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑝 → (2nd ‘((𝑡‘𝑗)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑝)))
161	160	cbvmptv 5279	. . . . . . . . . 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 2780	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝))))
164	163	cbvmptv 5279	. . . . . . 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 46549	. . . . . 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 583	. . . . 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 812	. . . 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 583	. . 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 46453	. 2 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
170	1	dmvon 46527	. . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
171	170	eqcomd 2746	. 2 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
172	169, 171	eleqtrd 2846	1 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ifcif 4548  𝒫 cpw 4622  ∪ cuni 4931  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  dom cdm 5700   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029   ↑_m cmap 8884  Xcixp 8955  Fincfn 9003  ℝcr 11183  0cc0 11184  +∞cpnf 11321  -∞cmnf 11322  ℝ^*cxr 11323   ≤ cle 11325  ℕcn 12293  ℝ⁺crp 13057   +_𝑒 cxad 13173  (,)cioo 13407  [,)cico 13409  [,]cicc 13410  ∏cprod 15951  volcvol 25517  Σ^{^}csumge0 46283  CaraGenccaragen 46412  voln*covoln 46457  volncvoln 46459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-acn 10011  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-prod 15952  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-rest 17482  df-0g 17501  df-topgen 17503  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-subg 19163  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-drng 20753  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-cnfld 21388  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518  df-vol 25519  df-sumge0 46284  df-ome 46411  df-caragen 46413  df-ovoln 46458  df-voln 46460

This theorem is referenced by:  hoimbllem  46551