hsphoidmvle2 - Mathbox for Glauco Siliprandi

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Theorem hsphoidmvle2 47013

Description: The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hsphoidmvle2.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hsphoidmvle2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hsphoidmvle2.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hsphoidmvle2.y	⊢ 𝑋 = (𝑌 ∪ {𝑍})
hsphoidmvle2.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
hsphoidmvle2.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
hsphoidmvle2.e	⊢ (𝜑 → 𝐶 ≤ 𝐷)
hsphoidmvle2.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hsphoidmvle2.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
hsphoidmvle2.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)

**Assertion**
Ref	Expression
hsphoidmvle2	⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)))

Distinct variable groups: 𝐴,𝑎,𝑏,𝑘 𝐵,𝑎,𝑏,𝑘 𝐵,𝑐,𝑗,𝑘 𝐶,𝑎,𝑏,𝑘,𝑥 𝐶,𝑐,𝑗,𝑥 𝐷,𝑎,𝑏,𝑘,𝑥 𝐷,𝑐,𝑗 𝐻,𝑎,𝑏,𝑘 𝑋,𝑎,𝑏,𝑘,𝑥 𝑋,𝑐,𝑗 𝑌,𝑐,𝑗,𝑥 𝑍,𝑐,𝑗,𝑘,𝑥 𝜑,𝑎,𝑏,𝑘,𝑥 𝜑,𝑐,𝑗 Allowed substitution hints: 𝐴(𝑥,𝑗,𝑐) 𝐵(𝑥) 𝐻(𝑥,𝑗,𝑐) 𝐿(𝑥,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑌(𝑘,𝑎,𝑏) 𝑍(𝑎,𝑏)

**Proof of Theorem hsphoidmvle2**
Step	Hyp	Ref	Expression
1		hsphoidmvle2.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
2		hsphoidmvle2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
3	2	eldifad 3901	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelcdmd 7037	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle2.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelcdmd 7037	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4513	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 47009	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
10	4, 8, 9	syl2anc 585	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
11		hsphoidmvle2.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
12	6, 11	ifcld 4513	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ)
13		volicore 47009	. . . 4 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ) → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) ∈ ℝ)
14	4, 12, 13	syl2anc 585	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) ∈ ℝ)
15		hsphoidmvle2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
16		difssd 4077	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17		ssfi 9107	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
18	15, 16, 17	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19		eldifi 4071	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
20	19	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
21	1	ffvelcdmda 7036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
22	5	ffvelcdmda 7036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
23		volicore 47009	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
24	21, 22, 23	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
25	20, 24	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
26	18, 25	fprodrecl 15918	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
27		nfv 1916	. . . 4 ⊢ Ⅎ𝑘𝜑
28	20, 21	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴‘𝑘) ∈ ℝ)
29	20, 22	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ)
30	29	rexrd 11195	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ^*)
31		icombl 25531	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
32	28, 30, 31	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
33		volge0 46389	. . . . 5 ⊢ (((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	32, 33	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
35	27, 18, 25, 34	fprodge0 15958	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
36	8	rexrd 11195	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ^*)
37		icombl 25531	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ^*) → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
38	4, 36, 37	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
39	12	rexrd 11195	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ^*)
40		icombl 25531	. . . . 5 ⊢ (((𝐴‘𝑍) ∈ ℝ ∧ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ^*) → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)) ∈ dom vol)
41	4, 39, 40	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)) ∈ dom vol)
42	4	rexrd 11195	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
43	4	leidd 11716	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
44	6	leidd 11716	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
45	44	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
46		iftrue 4472	. . . . . . . . 9 ⊢ ((𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
48	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ∈ ℝ)
49	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
50	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐶)
52		hsphoidmvle2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ≤ 𝐷)
54	48, 49, 50, 51, 53	letrd 11303	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐷)
55	54	iftrued 4474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
56	47, 55	breq12d 5098	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ↔ (𝐵‘𝑍) ≤ (𝐵‘𝑍)))
57	45, 56	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
58		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝜑)
59		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → ¬ (𝐵‘𝑍) ≤ 𝐶)
60	58, 7	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
61	58, 6	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ∈ ℝ)
62	60, 61	ltnled 11293	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → (𝐶 < (𝐵‘𝑍) ↔ ¬ (𝐵‘𝑍) ≤ 𝐶))
63	59, 62	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 < (𝐵‘𝑍))
64	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ∈ ℝ)
65	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → (𝐵‘𝑍) ∈ ℝ)
66		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 < (𝐵‘𝑍))
67	64, 65, 66	ltled 11294	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ≤ (𝐵‘𝑍))
68	67	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵‘𝑍))
69		iftrue 4472	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
70	69	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
71	70	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
72	68, 71	breqtrd 5111	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
73	52	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ 𝐷)
74		iffalse 4475	. . . . . . . . . . . 12 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = 𝐷)
75	74	eqcomd 2742	. . . . . . . . . . 11 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
76	75	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
77	73, 76	breqtrd 5111	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
78	72, 77	pm2.61dan 813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
79	58, 63, 78	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
80		iffalse 4475	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
81	80	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
82	81	breq1d 5095	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → (if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ↔ 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
83	79, 82	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
84	57, 83	pm2.61dan 813	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
85		icossico 13369	. . . . 5 ⊢ ((((𝐴‘𝑍) ∈ ℝ^* ∧ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ^*) ∧ ((𝐴‘𝑍) ≤ (𝐴‘𝑍) ∧ if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
86	42, 39, 43, 84, 85	syl22anc 839	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
87		volss 25500	. . . 4 ⊢ ((((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol ∧ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)) ∈ dom vol ∧ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))))
88	38, 41, 86, 87	syl3anc 1374	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ≤ (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))))
89	10, 14, 26, 35, 88	lemul1ad 12095	. 2 ⊢ (𝜑 → ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ≤ ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
90		hsphoidmvle2.l	. . . . 5 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
91	3	ne0d 4282	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
92		hsphoidmvle2.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
93	92, 7, 15, 5	hsphoif 47004	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
94	90, 15, 91, 1, 93	hoidmvn0val 47012	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
95	93	ffvelcdmda 7036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96		volicore 47009	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
97	21, 95, 96	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
98	97	recnd 11173	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
100		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
101	99, 100	oveq12d 7385	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
102	101	fveq2d 6844	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
103	102	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
104	92, 7, 15, 5, 3	hsphoival 47007	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
105	2	eldifbd 3902	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
106	105	iffalsed 4477	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
107	104, 106	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
108	107	oveq2d 7383	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
109	108	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
110	109	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
111	103, 110	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
112	15, 98, 3, 111	fprodsplit1 46023	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)))))
113	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
114	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
115	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
116	92, 113, 114, 115, 20	hsphoival 47007	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)))
117		hsphoidmvle2.y	. . . . . . . . . . . . 13 ⊢ 𝑋 = (𝑌 ∪ {𝑍})
118	19, 117	eleqtrdi 2846	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119		eldifn 4072	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120		elunnel2 4095	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
121	118, 119, 120	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
122	121	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
123	122	iftrued 4474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
124	116, 123	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
125	124	oveq2d 7383	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
126	125	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
127	126	prodeq2dv 15887	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
128	127	oveq2d 7383	. . . 4 ⊢ (𝜑 → ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)))) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
129	94, 112, 128	3eqtrd 2775	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
130	92, 11, 15, 5	hsphoif 47004	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐷)‘𝐵):𝑋⟶ℝ)
131	90, 15, 91, 1, 130	hoidmvn0val 47012	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))))
132	130	ffvelcdmda 7036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133		volicore 47009	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
134	21, 132, 133	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
135	134	recnd 11173	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136		fveq2 6840	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (((𝐻‘𝐷)‘𝐵)‘𝑍))
137	99, 136	oveq12d 7385	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)))
138	137	fveq2d 6844	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
139	138	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
140	15, 135, 3, 139	fprodsplit1 46023	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = ((vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)))))
141	92, 11, 15, 5, 3	hsphoival 47007	. . . . . . . 8 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
142	105	iffalsed 4477	. . . . . . . 8 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
143	141, 142	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
144	143	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
145	144	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))))
146	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
147	92, 146, 114, 115, 20	hsphoival 47007	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)))
148	122	iftrued 4474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)) = (𝐵‘𝑘))
149	147, 148	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (𝐵‘𝑘))
150	149	oveq2d 7383	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
151	150	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
152	151	prodeq2dv 15887	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
153	145, 152	oveq12d 7385	. . . 4 ⊢ (𝜑 → ((vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)))) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
154	131, 140, 153	3eqtrd 2775	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)) = ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))))
155	129, 154	breq12d 5098	. 2 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)) ↔ ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ≤ ((vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) · ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))))
156	89, 155	mpbird 257	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∖ cdif 3886 ∪ cun 3887 ⊆ wss 3889 ∅c0 4273 ifcif 4466 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ↑_m cmap 8773 Fincfn 8893 ℝcr 11037 0cc0 11038 · cmul 11043 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 [,)cico 13300 ∏cprod 15868 volcvol 25430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-prod 15869 df-rest 17385 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-bases 22911 df-cmp 23352 df-ovol 25431 df-vol 25432

This theorem is referenced by: hoidmvlelem1 47023 hoidmvlelem2 47024

W3C validator