hspmbl - Mathbox for Glauco Siliprandi

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Theorem hspmbl 45393

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 45337	. . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3		eqid 2733	. . 3 ⊢ ∪ dom (voln‘𝑋) = ∪ dom (voln‘𝑋)
4		eqid 2733	. . 3 ⊢ (CaraGen‘(voln‘𝑋)) = (CaraGen‘(voln‘𝑋))
5		ovex 7442	. . . . . . . . 9 ⊢ (-∞(,)𝑌) ∈ V
6		reex 11201	. . . . . . . . 9 ⊢ ℝ ∈ V
7	5, 6	ifex 4579	. . . . . . . 8 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
8	7	ixpssmap 8926	. . . . . . 7 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑_m 𝑋)
9		iftrue 4535	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10		ioossre 13385	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑌) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
12	9, 11	eqsstrd 4021	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13		iffalse 4538	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14		ssid 4005	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → ℝ ⊆ ℝ)
16	13, 15	eqsstrd 4021	. . . . . . . . . . 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 3066	. . . . . . . . 9 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19		iunss 5049	. . . . . . . . 9 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
20	18, 19	mpbir 230	. . . . . . . 8 ⊢ ∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21		mapss 8883	. . . . . . . 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 3992	. . . . . 6 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑_m 𝑋)
24	7	rgenw 3066	. . . . . . . 8 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25		ixpexg 8916	. . . . . . . 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 4606	. . . . . . 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 230	. . . . 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 2016	. . . . . . . . 9 ⊢ 𝑥 = 𝑥
33		eqid 2733	. . . . . . . . 9 ⊢ ℝ = ℝ
34		equequ1 2029	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑝 → (𝑘 = 𝑙 ↔ 𝑝 = 𝑙))
35	34	ifbid 4552	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
36	35	cbvixpv 8909	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
37	32, 33, 36	mpoeq123i 7485	. . . . . . . 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 5254	. . . . . . 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 2761	. . . . . 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 45373	. . . . 5 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) = X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
43	1	ovnf 45327	. . . . . . . . 9 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))
44	43	fdmd 6729	. . . . . . . 8 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑_m 𝑋))
45	44	unieqd 4923	. . . . . . 7 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑_m 𝑋))
46		unipw 5451	. . . . . . . 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 2773	. . . . . 6 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑_m 𝑋))
49	48	pweqd 4620	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑_m 𝑋))
50	42, 49	eleq12d 2828	. . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝜑)
53		simpr 486	. . . . 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 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋))
56	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
57		inss1 4229	. . . . . . . . . . . . 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 4610	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) → 𝑎 ⊆ (ℝ ↑_m 𝑋))
60	58, 59	sstrd 3993	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
61	60	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
62	56, 61	ovnxrcl 45333	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → ((voln‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ^)
63	59	adantl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑎 ⊆ (ℝ ↑_m 𝑋))
64	63	ssdifssd 4143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → (𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
65	56, 64	ovnxrcl 45333	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → ((voln‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ^)
66	62, 65	xaddcld 13280	. . . . . . . 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 13110	. . . . . . . 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 482	. . . . . 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 2739	. . . . . . 7 ⊢ (((voln‘𝑋)‘𝑎) = +∞ → +∞ = ((voln‘𝑋)‘𝑎))
72	71	adantl 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ((voln‘𝑋)‘𝑎) = +∞) → +∞ = ((voln‘𝑋)‘𝑎))
73	69, 72	breqtrd 5175	. . . . 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 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)))
75	56, 63	ovncl 45331	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
76	75	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ¬ ((voln‘𝑋)‘𝑎) = +∞) → ((voln‘𝑋)‘𝑎) ∈ (0[,]+∞))
77		neqne 2949	. . . . . . . 8 ⊢ (¬ ((voln‘𝑋)‘𝑎) = +∞ → ((voln‘𝑋)‘𝑎) ≠ +∞)
78	77	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ¬ ((voln‘𝑋)‘𝑎) = +∞) → ((voln‘𝑋)‘𝑎) ≠ +∞)
79		ge0xrre 44292	. . . . . . 7 ⊢ ((((voln‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
80	76, 78, 79	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ¬ ((voln‘𝑋)‘𝑎) = +∞) → ((voln‘𝑋)‘𝑎) ∈ ℝ)
81	56	adantr 482	. . . . . . 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 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ((voln‘𝑋)‘𝑎) ∈ ℝ) → ((voln‘𝑋)‘𝑎) ∈ ℝ)
85	63	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑_m 𝑋))
86		sseq1 4008	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
87	86	rabbidv 3441	. . . . . . . 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 5262	. . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → 𝑖 = ℎ)
90	89	coeq2d 5863	. . . . . . . . . . 11 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ℎ))
91	90	fveq1d 6894	. . . . . . . . . 10 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ ℎ)‘𝑝))
92	91	fveq2d 6896	. . . . . . . . 9 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ ℎ)‘𝑝)))
93	92	prodeq2dv 15867	. . . . . . . 8 ⊢ (𝑖 = ℎ → ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
94	93	cbvmptv 5262	. . . . . . 7 ⊢ (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
95		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑝 → (([,) ∘ (𝑚‘𝑖))‘𝑛) = (([,) ∘ (𝑚‘𝑖))‘𝑝))
96	95	cbvixpv 8909	. . . . . . . . . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = ℎ → (𝑚‘𝑖) = (ℎ‘𝑖))
99	98	coeq2d 5863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ℎ → ([,) ∘ (𝑚‘𝑖)) = ([,) ∘ (ℎ‘𝑖)))
100	99	fveq1d 6894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ℎ → (([,) ∘ (𝑚‘𝑖))‘𝑝) = (([,) ∘ (ℎ‘𝑖))‘𝑝))
101	100	ixpeq2dv 8907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ℎ → X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
102	97, 101	eqtrd 2773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = ℎ → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
103	102	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = ℎ ∧ 𝑖 ∈ ℕ) → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
104	103	iuneq2dv 5022	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = ℎ → ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
105	104	sseq2d 4015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = ℎ → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) ↔ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)))
106	105	cbvrabv 3443	. . . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑙 → (ℎ‘𝑖) = (𝑙‘𝑖))
108	107	coeq2d 5863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑙 → ([,) ∘ (ℎ‘𝑖)) = ([,) ∘ (𝑙‘𝑖)))
109	108	fveq1d 6894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑙 → (([,) ∘ (ℎ‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑖))‘𝑝))
110	109	ixpeq2dv 8907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑙 → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
111	110	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = 𝑙 ∧ 𝑖 ∈ ℕ) → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
112	111	iuneq2dv 5022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
113		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (𝑙‘𝑖) = (𝑙‘𝑗))
114	113	coeq2d 5863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → ([,) ∘ (𝑙‘𝑖)) = ([,) ∘ (𝑙‘𝑗)))
115	114	fveq1d 6894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑗 → (([,) ∘ (𝑙‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑗))‘𝑝))
116	115	ixpeq2dv 8907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
117	116	cbviunv 5044	. . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
120	119	sseq2d 4015	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑙 → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
121	120	cbvrabv 3443	. . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . 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 5254	. . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . 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 6897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
129	128	cbvprodv 15860	. . . . . . . . . . . . . . . . . . 19 ⊢ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
130	129	mpteq2i 5254	. . . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑡‘𝑚) = (𝑡‘𝑗))
133	131, 132	fveq12d 6899	. . . . . . . . . . . . . . . 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 5262	. . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((voln‘𝑋)‘𝑐) = ((voln‘𝑋)‘𝑏))
138	137	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((voln‘𝑋)‘𝑐) +_𝑒 𝑠) = (((voln‘𝑋)‘𝑏) +_𝑒 𝑠))
139	136, 138	breq12d 5162	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 3435	. . . . . . . . . 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 5251	. . . . . . . . 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 2734	. . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑟 → (((voln‘𝑋)‘𝑏) +_𝑒 𝑠) = (((voln‘𝑋)‘𝑏) +_𝑒 𝑟))
146	145	breq2d 5161	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 3435	. . . . . . . . . . 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 5262	. . . . . . . . . 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 2773	. . . . . . . 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 5262	. . . . . . 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 6897	. . . . . . . . 9 ⊢ (𝑚 = 𝑝 → (1^st ‘((𝑡‘𝑗)‘𝑚)) = (1^st ‘((𝑡‘𝑗)‘𝑝)))
154	153	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑋 ↦ (1^st ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (1^st ‘((𝑡‘𝑗)‘𝑝)))
155	154	mpteq2i 5254	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (1^st ‘((𝑡‘𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (1^st ‘((𝑡‘𝑗)‘𝑝))))
156		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑡‘𝑖) = (𝑡‘𝑗))
157	156	fveq1d 6894	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑡‘𝑖)‘𝑚) = ((𝑡‘𝑗)‘𝑚))
158	157	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (2^nd ‘((𝑡‘𝑖)‘𝑚)) = (2^nd ‘((𝑡‘𝑗)‘𝑚)))
159	158	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑚))))
160		2fveq3 6897	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑝 → (2^nd ‘((𝑡‘𝑗)‘𝑚)) = (2^nd ‘((𝑡‘𝑗)‘𝑝)))
161	160	cbvmptv 5262	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑝)))
162	161	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑝))))
163	159, 162	eqtrd 2773	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑝))))
164	163	cbvmptv 5262	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (2^nd ‘((𝑡‘𝑗)‘𝑝))))
165	39, 81, 82, 83, 84, 85, 88, 94, 152, 155, 164	hspmbllem3 45392	. . . . . 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 585	. . . . 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 585	. . 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 45296	. 2 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
170	1	dmvon 45370	. . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
171	170	eqcomd 2739	. 2 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
172	169, 171	eleqtrd 2836	1 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ifcif 4529 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 ∘ ccom 5681 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 Xcixp 8891 Fincfn 8939 ℝcr 11109 0cc0 11110 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 ≤ cle 11249 ℕcn 12212 ℝ⁺crp 12974 +_𝑒 cxad 13090 (,)cioo 13324 [,)cico 13326 [,]cicc 13327 ∏cprod 15849 volcvol 24980 Σ^{^}csumge0 45126 CaraGenccaragen 45255 voln*covoln 45300 volncvoln 45302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-prod 15850 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-subg 19003 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-drng 20359 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982 df-sumge0 45127 df-ome 45254 df-caragen 45256 df-ovoln 45301 df-voln 45303

This theorem is referenced by: hoimbllem 45394

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