hsphoidmvle2 - Mathbox for Glauco Siliprandi

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Theorem hsphoidmvle2 47868

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 4054	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
4	1, 3	ffvelcdmd 7117	. . . 4 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
5		hsphoidmvle2.b	. . . . . 6 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
6	5, 3	ffvelcdmd 7117	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
7		hsphoidmvle2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
8	6, 7	ifcld 4532	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ)
9		volicore 47864	. . . 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 596	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))) ∈ ℝ)
11		hsphoidmvle2.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
12	6, 11	ifcld 4532	. . . 4 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ)
13		volicore 47864	. . . 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 596	. . 3 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))) ∈ ℝ)
15		hsphoidmvle2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
16		difssd 4076	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ⊆ 𝑋)
17		ssfi 9226	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → (𝑋 ∖ {𝑍}) ∈ Fin)
18	15, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑋 ∖ {𝑍}) ∈ Fin)
19		eldifi 4070	. . . . . 6 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑋)
20	19	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑋)
21	1	ffvelcdmda 7116	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
22	5	ffvelcdmda 7116	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
23		volicore 47864	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
24	21, 22, 23	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
25	20, 24	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
26	18, 25	fprodrecl 16164	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
27		nfv 1949	. . . 4 ⊢ Ⅎ𝑘𝜑
28	20, 21	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐴‘𝑘) ∈ ℝ)
29	20, 22	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ)
30	29	rexrd 11398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (𝐵‘𝑘) ∈ ℝ^*)
31		icombl 25940	. . . . . 6 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ^*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
32	28, 30, 31	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol)
33		volge0 47244	. . . . 5 ⊢ (((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
34	32, 33	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
35	27, 18, 25, 34	fprodge0 16204	. . 3 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
36	8	rexrd 11398	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ∈ ℝ^*)
37		icombl 25940	. . . . 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 596	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ∈ dom vol)
39	12	rexrd 11398	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) ∈ ℝ^*)
40		icombl 25940	. . . . 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 596	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)) ∈ dom vol)
42	4	rexrd 11398	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
43	4	leidd 11922	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
44	6	leidd 11922	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
45	44	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ (𝐵‘𝑍))
46		iftrue 4491	. . . . . . . . 9 ⊢ ((𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
47	46	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = (𝐵‘𝑍))
48	6	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ∈ ℝ)
49	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
50	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐷 ∈ ℝ)
51		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐶)
52		hsphoidmvle2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
53	52	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ≤ 𝐷)
54	48, 49, 50, 51, 53	letrd 11506	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ≤ 𝐷)
55	54	iftrued 4493	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
56	47, 55	breq12d 5121	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
58		simpl 488	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝜑)
59		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → ¬ (𝐵‘𝑍) ≤ 𝐶)
60	58, 7	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ∈ ℝ)
61	58, 6	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → (𝐵‘𝑍) ∈ ℝ)
62	60, 61	ltnled 11496	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → (𝐶 < (𝐵‘𝑍) ↔ ¬ (𝐵‘𝑍) ≤ 𝐶))
63	59, 62	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 < (𝐵‘𝑍))
64	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ∈ ℝ)
65	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → (𝐵‘𝑍) ∈ ℝ)
66		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 < (𝐵‘𝑍))
67	64, 65, 66	ltled 11497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ≤ (𝐵‘𝑍))
68	67	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ (𝐵‘𝑍))
69		iftrue 4491	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = (𝐵‘𝑍))
70	69	eqcomd 2769	. . . . . . . . . . 11 ⊢ ((𝐵‘𝑍) ≤ 𝐷 → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
71	70	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → (𝐵‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
72	68, 71	breqtrd 5137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
73	52	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ 𝐷)
74		iffalse 4494	. . . . . . . . . . . 12 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷) = 𝐷)
75	74	eqcomd 2769	. . . . . . . . . . 11 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐷 → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
76	75	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐷 = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
77	73, 76	breqtrd 5137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) ∧ ¬ (𝐵‘𝑍) ≤ 𝐷) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
78	72, 77	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < (𝐵‘𝑍)) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
79	58, 63, 78	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → 𝐶 ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
80		iffalse 4494	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑍) ≤ 𝐶 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
81	80	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) = 𝐶)
82	81	breq1d 5118	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝐶) → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
84	57, 83	pm2.61dan 825	. . . . 5 ⊢ (𝜑 → if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶) ≤ if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
85		icossico 13587	. . . . 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 852	. . . 4 ⊢ (𝜑 → ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)) ⊆ ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
87		volss 25909	. . . 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 1399	. . 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 12296	. 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 4291	. . . . 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 47859	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐶)‘𝐵):𝑋⟶ℝ)
94	90, 15, 91, 1, 93	hoidmvn0val 47867	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))))
95	93	ffvelcdmda 7116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ)
96		volicore 47864	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐶)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
97	21, 95, 96	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℝ)
98	97	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) ∈ ℂ)
99		fveq2 6916	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
100		fveq2 6916	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (((𝐻‘𝐶)‘𝐵)‘𝑍))
101	99, 100	oveq12d 7469	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)))
102	101	fveq2d 6920	. . . . . . 7 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))))
103	102	adantl 487	. . . . . 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 47862	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
105	2	eldifbd 4055	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
106	105	iffalsed 4496	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (𝜑 → (((𝐻‘𝐶)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))
108	107	oveq2d 7467	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶)))
109	108	fveq2d 6920	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐶)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐶, (𝐵‘𝑍), 𝐶))))
110	109	adantr 486	. . . . . 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 2798	. . . . 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 46878	. . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐶 ∈ ℝ)
114	15	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑋 ∈ Fin)
115	5	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐵:𝑋⟶ℝ)
116	92, 113, 114, 115, 20	hsphoival 47862	. . . . . . . . 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 2874	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
119		eldifn 4071	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → ¬ 𝑘 ∈ {𝑍})
120		elunnel2 4157	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ {𝑍}) → 𝑘 ∈ 𝑌)
121	118, 119, 120	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑋 ∖ {𝑍}) → 𝑘 ∈ 𝑌)
122	121	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝑘 ∈ 𝑌)
123	122	iftrued 4493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐶, (𝐵‘𝑘), 𝐶)) = (𝐵‘𝑘))
124	116, 123	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐶)‘𝐵)‘𝑘) = (𝐵‘𝑘))
125	124	oveq2d 7467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
126	125	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
127	126	prodeq2dv 16134	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐶)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
128	127	oveq2d 7467	. . . 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 2802	. . 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 47859	. . . . 5 ⊢ (𝜑 → ((𝐻‘𝐷)‘𝐵):𝑋⟶ℝ)
131	90, 15, 91, 1, 130	hoidmvn0val 47867	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))))
132	130	ffvelcdmda 7116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (((𝐻‘𝐷)‘𝐵)‘𝑘) ∈ ℝ)
133		volicore 47864	. . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (((𝐻‘𝐷)‘𝐵)‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
134	21, 132, 133	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℝ)
135	134	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) ∈ ℂ)
136		fveq2 6916	. . . . . . . 8 ⊢ (𝑘 = 𝑍 → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (((𝐻‘𝐷)‘𝐵)‘𝑍))
137	99, 136	oveq12d 7469	. . . . . . 7 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)))
138	137	fveq2d 6920	. . . . . 6 ⊢ (𝑘 = 𝑍 → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
139	138	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))))
140	15, 135, 3, 139	fprodsplit1 46878	. . . 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 47862	. . . . . . . 8 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if(𝑍 ∈ 𝑌, (𝐵‘𝑍), if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
142	105	iffalsed 4496	. . . . . . . 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 2798	. . . . . . 7 ⊢ (𝜑 → (((𝐻‘𝐷)‘𝐵)‘𝑍) = if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))
144	143	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍)) = ((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷)))
145	144	fveq2d 6920	. . . . 5 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(((𝐻‘𝐷)‘𝐵)‘𝑍))) = (vol‘((𝐴‘𝑍)[,)if((𝐵‘𝑍) ≤ 𝐷, (𝐵‘𝑍), 𝐷))))
146	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → 𝐷 ∈ ℝ)
147	92, 146, 114, 115, 20	hsphoival 47862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)))
148	122	iftrued 4493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → if(𝑘 ∈ 𝑌, (𝐵‘𝑘), if((𝐵‘𝑘) ≤ 𝐷, (𝐵‘𝑘), 𝐷)) = (𝐵‘𝑘))
149	147, 148	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (((𝐻‘𝐷)‘𝐵)‘𝑘) = (𝐵‘𝑘))
150	149	oveq2d 7467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → ((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
151	150	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑋 ∖ {𝑍})) → (vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
152	151	prodeq2dv 16134	. . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(((𝐻‘𝐷)‘𝐵)‘𝑘))) = ∏𝑘 ∈ (𝑋 ∖ {𝑍})(vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
153	145, 152	oveq12d 7469	. . . 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 2802	. . 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 5121	. 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 260	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)((𝐻‘𝐶)‘𝐵)) ≤ (𝐴(𝐿‘𝑋)((𝐻‘𝐷)‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 ∅c0 4281 ifcif 4485 {csn 4588 class class class wbr 5108 ↦ cmpt 5192 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 ↑_m cmap 8885 Fincfn 9011 ℝcr 11238 0cc0 11239 · cmul 11244 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 [,)cico 13518 ∏cprod 16116 volcvol 25839

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-rlim 15700 df-sum 15898 df-prod 16117 df-rest 17643 df-topgen 17664 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-top 23267 df-topon 23284 df-bases 23319 df-cmp 23760 df-ovol 25840 df-vol 25841

This theorem is used by: hoidmvlelem1 47878 hoidmvlelem2 47879

W3C validator