hsphoidmvle2 - Mathbox for Glauco Siliprandi

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Theorem hsphoidmvle2 45236

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 3959	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelcdmd 7083	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle2.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelcdmd 7083	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4573	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 45232	. . . 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 4573	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ)
13		volicore 45232	. . . 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 4131	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17		ssfi 9169	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
18	15, 16, 17	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19		eldifi 4125	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
20	19	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
21	1	ffvelcdmda 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
22	5	ffvelcdmda 7082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
23		volicore 45232	. . . . . 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 15893	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
27		nfv 1918	. . . 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 11260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ^*)
31		icombl 25063	. . . . . 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 44612	. . . . 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 15933	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
36	8	rexrd 11260	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ^*)
37		icombl 25063	. . . . 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 11260	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ^*)
40		icombl 25063	. . . . 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 11260	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
43	4	leidd 11776	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
44	6	leidd 11776	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
45	44	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
46		iftrue 4533	. . . . . . . . 9 ⊢ ((𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
47	46	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
48	6	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ∈ ℝ)
49	7	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
50	11	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐶)
52		hsphoidmvle2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
53	52	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ≤ 𝐷)
54	48, 49, 50, 51, 53	letrd 11367	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐷)
55	54	iftrued 4535	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
56	47, 55	breq12d 5160	. . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝜑)
59		simpr 486	. . . . . . . . 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 11357	. . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ∈ ℝ)
65	6	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → (𝐵‘𝑍) ∈ ℝ)
66		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 < (𝐵‘𝑍))
67	64, 65, 66	ltled 11358	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ≤ (𝐵‘𝑍))
68	67	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵‘𝑍))
69		iftrue 4533	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
70	69	eqcomd 2739	. . . . . . . . . . 11 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
71	70	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
72	68, 71	breqtrd 5173	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
73	52	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ 𝐷)
74		iffalse 4536	. . . . . . . . . . . 12 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = 𝐷)
75	74	eqcomd 2739	. . . . . . . . . . 11 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
76	75	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
77	73, 76	breqtrd 5173	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
78	72, 77	pm2.61dan 812	. . . . . . . 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 4536	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
81	80	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
82	81	breq1d 5157	. . . . . . 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 812	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
85		icossico 13390	. . . . 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 838	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
87		volss 25032	. . . 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 1372	. . 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 12149	. 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 4334	. . . . 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 45227	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
94	90, 15, 91, 1, 93	hoidmvn0val 45235	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
95	93	ffvelcdmda 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96		volicore 45232	. . . . . . 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 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99		fveq2 6888	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
100		fveq2 6888	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
101	99, 100	oveq12d 7422	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
102	101	fveq2d 6892	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
103	102	adantl 483	. . . . . 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 45230	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
105	2	eldifbd 3960	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
106	105	iffalsed 4538	. . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
108	107	oveq2d 7420	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
109	108	fveq2d 6892	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
110	109	adantr 482	. . . . . 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 2773	. . . . 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 44244	. . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
114	15	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
115	5	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
116	92, 113, 114, 115, 20	hsphoival 45230	. . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119		eldifn 4126	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120		elunnel2 4149	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
121	118, 119, 120	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
122	121	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
123	122	iftrued 4535	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
124	116, 123	eqtrd 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
125	124	oveq2d 7420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
126	125	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
127	126	prodeq2dv 15863	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
128	127	oveq2d 7420	. . . 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 2777	. . 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 45227	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐷)‘𝐵):𝑋⟶ℝ)
131	90, 15, 91, 1, 130	hoidmvn0val 45235	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))))
132	130	ffvelcdmda 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133		volicore 45232	. . . . . . 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 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136		fveq2 6888	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (((𝐻‘𝐷)‘𝐵)‘𝑍))
137	99, 136	oveq12d 7422	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)))
138	137	fveq2d 6892	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
139	138	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
140	15, 135, 3, 139	fprodsplit1 44244	. . . 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 45230	. . . . . . . 8 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
142	105	iffalsed 4538	. . . . . . . 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 2773	. . . . . . 7 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
144	143	oveq2d 7420	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
145	144	fveq2d 6892	. . . . 5 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))))
146	11	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
147	92, 146, 114, 115, 20	hsphoival 45230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)))
148	122	iftrued 4535	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)) = (𝐵‘𝑘))
149	147, 148	eqtrd 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (𝐵‘𝑘))
150	149	oveq2d 7420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
151	150	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
152	151	prodeq2dv 15863	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
153	145, 152	oveq12d 7422	. . . 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 2777	. . 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 5160	. 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 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 ifcif 4527 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 ↑_m cmap 8816 Fincfn 8935 ℝcr 11105 0cc0 11106 · cmul 11111 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 ∏cprod 15845 volcvol 24962

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-prod 15846 df-rest 17364 df-topgen 17385 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-top 22378 df-topon 22395 df-bases 22431 df-cmp 22873 df-ovol 24963 df-vol 24964

This theorem is referenced by: hoidmvlelem1 45246 hoidmvlelem2 45247

W3C validator