Theorem hspmbl 42904

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 42848	. . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3		eqid 2821	. . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋)
4		eqid 2821	. . 3 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5		ovex 7183	. . . . . . . . 9 ⊢ (-∞(,)𝑌) ∈ V
6		reex 10622	. . . . . . . . 9 ⊢ ℝ ∈ V
7	5, 6	ifex 4515	. . . . . . . 8 ⊢ if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
8	7	ixpssmap 8490	. . . . . . 7 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋)
9		iftrue 4473	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10		ioossre 12792	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑌) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
12	9, 11	eqsstrd 4005	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13		iffalse 4476	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14		ssid 3989	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ 𝑝 = 𝐾 → ℝ ⊆ ℝ)
16	13, 15	eqsstrd 4005	. . . . . . . . . . 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 3150	. . . . . . . . 9 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19		iunss 4962	. . . . . . . . 9 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
20	18, 19	mpbir 233	. . . . . . . 8 ⊢ ∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21		mapss 8447	. . . . . . . 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 690	. . . . . . 7 ⊢ (∪ 𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)
23	8, 22	sstri 3976	. . . . . 6 ⊢ X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)
24	7	rgenw 3150	. . . . . . . 8 ⊢ ∀𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25		ixpexg 8480	. . . . . . . 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 4545	. . . . . . 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 233	. . . . 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 2015	. . . . . . . . 9 ⊢ 𝑥 = 𝑥
33		eqid 2821	. . . . . . . . 9 ⊢ ℝ = ℝ
34		equequ1 2028	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑝 → (𝑘 = 𝑙 ↔ 𝑝 = 𝑙))
35	34	ifbid 4489	. . . . . . . . . 10 ⊢ (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
36	35	cbvixpv 8473	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝 ∈ 𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
37	32, 33, 36	mpoeq123i 7224	. . . . . . . 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 5151	. . . . . . 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 2844	. . . . . 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 42884	. . . . 5 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) = X𝑝 ∈ 𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
43	1	ovnf 42838	. . . . . . . . 9 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞))
44	43	fdmd 6518	. . . . . . . 8 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
45	44	unieqd 4842	. . . . . . 7 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑m 𝑋))
46		unipw 5335	. . . . . . . 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 2856	. . . . . 6 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋))
49	48	pweqd 4544	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
50	42, 49	eleq12d 2907	. . . 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 259	. . 3 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ 𝒫 ∪ dom (voln*‘𝑋))
52		simpl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝜑)
53		simpr 487	. . . . 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 2915	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋))
56	1	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑋 ∈ Fin)
57		inss1 4205	. . . . . . . . . . . . 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 4551	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → 𝑎 ⊆ (ℝ ↑m 𝑋))
60	58, 59	sstrd 3977	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
61	60	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
62	56, 61	ovnxrcl 42844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
63	59	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑎 ⊆ (ℝ ↑m 𝑋))
64	63	ssdifssd 4119	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
65	56, 64	ovnxrcl 42844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ*)
66	62, 65	xaddcld 12688	. . . . . . . 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 12519	. . . . . . . 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 483	. . . . . 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 2827	. . . . . . 7 ⊢ (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
72	71	adantl 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
73	69, 72	breqtrd 5085	. . . . 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 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)))
75	56, 63	ovncl 42842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
76	75	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
77		neqne 3024	. . . . . . . 8 ⊢ (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
78	77	adantl 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
79		ge0xrre 41799	. . . . . . 7 ⊢ ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
80	76, 78, 79	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
81	56	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
82	40	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾 ∈ 𝑋)
83	41	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
84		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
85	63	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑m 𝑋))
86		sseq1 3992	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
87	86	rabbidv 3481	. . . . . . . 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 5162	. . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → 𝑖 = ℎ)
90	89	coeq2d 5728	. . . . . . . . . . 11 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ℎ))
91	90	fveq1d 6667	. . . . . . . . . 10 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ ℎ)‘𝑝))
92	91	fveq2d 6669	. . . . . . . . 9 ⊢ ((𝑖 = ℎ ∧ 𝑝 ∈ 𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ ℎ)‘𝑝)))
93	92	prodeq2dv 15271	. . . . . . . 8 ⊢ (𝑖 = ℎ → ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
94	93	cbvmptv 5162	. . . . . . 7 ⊢ (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑝)))
95		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑝 → (([,) ∘ (𝑚‘𝑖))‘𝑛) = (([,) ∘ (𝑚‘𝑖))‘𝑝))
96	95	cbvixpv 8473	. . . . . . . . . . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = ℎ → (𝑚‘𝑖) = (ℎ‘𝑖))
99	98	coeq2d 5728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ℎ → ([,) ∘ (𝑚‘𝑖)) = ([,) ∘ (ℎ‘𝑖)))
100	99	fveq1d 6667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ℎ → (([,) ∘ (𝑚‘𝑖))‘𝑝) = (([,) ∘ (ℎ‘𝑖))‘𝑝))
101	100	ixpeq2dv 8471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ℎ → X𝑝 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
102	97, 101	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = ℎ → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
103	102	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = ℎ ∧ 𝑖 ∈ ℕ) → X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
104	103	iuneq2dv 4936	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = ℎ → ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝))
105	104	sseq2d 3999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = ℎ → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑛 ∈ 𝑋 (([,) ∘ (𝑚‘𝑖))‘𝑛) ↔ 𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝)))
106	105	cbvrabv 3492	. . . . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = 𝑙 → (ℎ‘𝑖) = (𝑙‘𝑖))
108	107	coeq2d 5728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑙 → ([,) ∘ (ℎ‘𝑖)) = ([,) ∘ (𝑙‘𝑖)))
109	108	fveq1d 6667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑙 → (([,) ∘ (ℎ‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑖))‘𝑝))
110	109	ixpeq2dv 8471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑙 → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
111	110	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ = 𝑙 ∧ 𝑖 ∈ ℕ) → X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
112	111	iuneq2dv 4936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝))
113		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑗 → (𝑙‘𝑖) = (𝑙‘𝑗))
114	113	coeq2d 5728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑗 → ([,) ∘ (𝑙‘𝑖)) = ([,) ∘ (𝑙‘𝑗)))
115	114	fveq1d 6667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑗 → (([,) ∘ (𝑙‘𝑖))‘𝑝) = (([,) ∘ (𝑙‘𝑗))‘𝑝))
116	115	ixpeq2dv 8471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑖))‘𝑝) = X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
117	116	cbviunv 4958	. . . . . . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑙 → ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) = ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝))
120	119	sseq2d 3999	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝑙 → (𝑎 ⊆ ∪ 𝑖 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (ℎ‘𝑖))‘𝑝) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑝 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑝)))
121	120	cbvrabv 3492	. . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . 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 2898	. . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
129	128	cbvprodv 15264	. . . . . . . . . . . . . . . . . . 19 ⊢ ∏𝑚 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
130	129	mpteq2i 5151	. . . . . . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑡‘𝑚) = (𝑡‘𝑗))
133	131, 132	fveq12d 6672	. . . . . . . . . . . . . . . 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 5162	. . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
138	137	oveq1d 7165	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
139	136, 138	breq12d 5072	. . . . . . . . . . . 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 3477	. . . . . . . . . 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 5155	. . . . . . . . 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 2822	. . . . . . . . . . . . . 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 2898	. . . . . . . . . . . . 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 7158	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
146	145	breq2d 5071	. . . . . . . . . . . . 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 3477	. . . . . . . . . . 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 5162	. . . . . . . . . 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 2856	. . . . . . . 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 5162	. . . . . . 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 6670	. . . . . . . . 9 ⊢ (𝑚 = 𝑝 → (1st ‘((𝑡‘𝑗)‘𝑚)) = (1st ‘((𝑡‘𝑗)‘𝑝)))
154	153	cbvmptv 5162	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝)))
155	154	mpteq2i 5151	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝑚 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝 ∈ 𝑋 ↦ (1st ‘((𝑡‘𝑗)‘𝑝))))
156		fveq2 6665	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑡‘𝑖) = (𝑡‘𝑗))
157	156	fveq1d 6667	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑡‘𝑖)‘𝑚) = ((𝑡‘𝑗)‘𝑚))
158	157	fveq2d 6669	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (2nd ‘((𝑡‘𝑖)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑚)))
159	158	mpteq2dv 5155	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑚))))
160		2fveq3 6670	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑝 → (2nd ‘((𝑡‘𝑗)‘𝑚)) = (2nd ‘((𝑡‘𝑗)‘𝑝)))
161	160	cbvmptv 5162	. . . . . . . . . 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 2856	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑚 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑖)‘𝑚))) = (𝑝 ∈ 𝑋 ↦ (2nd ‘((𝑡‘𝑗)‘𝑝))))
164	163	cbvmptv 5162	. . . . . . 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 42903	. . . . . 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 586	. . . . 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 811	. . . 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 586	. . 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 42807	. 2 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
170	1	dmvon 42881	. . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
171	170	eqcomd 2827	. 2 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
172	169, 171	eleqtrd 2915	1 ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {crab 3142  Vcvv 3495   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ifcif 4467  𝒫 cpw 4539  ∪ cuni 4832  ∪ ciun 4912   class class class wbr 5059   ↦ cmpt 5139   × cxp 5548  dom cdm 5550   ∘ ccom 5554  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  1^st c1st 7681  2^nd c2nd 7682   ↑_m cmap 8400  Xcixp 8455  Fincfn 8503  ℝcr 10530  0cc0 10531  +∞cpnf 10666  -∞cmnf 10667  ℝ^*cxr 10668   ≤ cle 10670  ℕcn 11632  ℝ⁺crp 12383   +_𝑒 cxad 12499  (,)cioo 12732  [,)cico 12734  [,]cicc 12735  ∏cprod 15253  volcvol 24058  Σ^{^}csumge0 42637  CaraGenccaragen 42766  voln*covoln 42811  volncvoln 42813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cc 9851  ax-ac2 9879  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-disj 5025  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-acn 9365  df-ac 9536  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-prod 15254  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-starv 16574  df-tset 16578  df-ple 16579  df-ds 16581  df-unif 16582  df-rest 16690  df-0g 16709  df-topgen 16711  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-subg 18270  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-ring 19293  df-cring 19294  df-oppr 19367  df-dvdsr 19385  df-unit 19386  df-invr 19416  df-dvr 19427  df-drng 19498  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-cnfld 20540  df-top 21496  df-topon 21513  df-bases 21548  df-cmp 21989  df-ovol 24059  df-vol 24060  df-sumge0 42638  df-ome 42765  df-caragen 42767  df-ovoln 42812  df-voln 42814

This theorem is referenced by:  hoimbllem  42905