Theorem ovnsubadd2lem 39535

Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovnsubadd2lem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnsubadd2lem.a	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
ovnsubadd2lem.b	⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋))
ovnsubadd2lem.c	⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))

Assertion

Ref	Expression
ovnsubadd2lem	⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑋   𝜑,𝑛

Proof of Theorem ovnsubadd2lem

Step	Hyp	Ref	Expression
1		ovnsubadd2lem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		iftrue 4042	. . . . . . . 8 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4		ovex 6577	. . . . . . . . . . 11 ⊢ (ℝ ↑𝑚 𝑋) ∈ V
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
6		ovnsubadd2lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
7	5, 6	ssexd 4733	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
8	7, 6	elpwd 38264	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
10	3, 9	eqeltrd 2688	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
11	10	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
12		simpl 472	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
13	12	iffalsed 4047	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
14		simpr 476	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
15	14	iftrued 4044	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
16	13, 15	eqtrd 2644	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
17	16	adantll 746	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
18		ovnsubadd2lem.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋))
19	5, 18	ssexd 4733	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
20	19, 18	elpwd 38264	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
21	20	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
22	17, 21	eqeltrd 2688	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
23	22	adantllr 751	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
24		simpl 472	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
25	24	iffalsed 4047	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
26		simpr 476	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
27	26	iffalsed 4047	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
28	25, 27	eqtrd 2644	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
29		0elpw 4760	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋)
30	29	a1i 11	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋))
31	28, 30	eqeltrd 2688	. . . . . . 7 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
32	31	adantll 746	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
33	23, 32	pm2.61dan 828	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
34	11, 33	pm2.61dan 828	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
35		ovnsubadd2lem.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
36	34, 35	fmptd 6292	. . 3 ⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
37	1, 36	ovnsubadd 39462	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
38		eldifi 3694	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
39	38	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
40		eldifn 3695	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
41		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑛 ∈ V
42	41	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
43		id 22	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
44	42, 43	nelpr1 38289	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
45	44	neneqd 2787	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
46	40, 45	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
47	42, 43	nelpr2 38288	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
48	47	neneqd 2787	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
49	40, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
50	46, 49, 28	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
51		0ex 4718	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
53	50, 52	eqeltrd 2688	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
54	53	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
55	35	fvmpt2 6200	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
56	39, 54, 55	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
57	50	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
58	56, 57	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = ∅)
59	58	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅)
60		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑛(ℕ ∖ {1, 2})
61	60	iunxdif3 4542	. . . . . . 7 ⊢ (∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅ → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
62	59, 61	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
63	62	eqcomd 2616	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛))
64		1nn 10908	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
65		2nn 11062	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
66	64, 65	pm3.2i 470	. . . . . . . . 9 ⊢ (1 ∈ ℕ ∧ 2 ∈ ℕ)
67		prssi 4293	. . . . . . . . 9 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ {1, 2} ⊆ ℕ
69		dfss4 3820	. . . . . . . 8 ⊢ ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
70	68, 69	mpbi 219	. . . . . . 7 ⊢ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
71		iuneq1 4470	. . . . . . 7 ⊢ ((ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2} → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛))
72	70, 71	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛)
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛))
74		fveq2 6103	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
75		fveq2 6103	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2))
76	74, 75	iunxprg 4543	. . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
77	64, 65, 76	mp2an 704	. . . . . . 7 ⊢ ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2))
78	77	a1i 11	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
79	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))))
80	64	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℕ)
81	79, 3, 80, 7	fvmptd 6197	. . . . . . 7 ⊢ (𝜑 → (𝐶‘1) = 𝐴)
82		id 22	. . . . . . . . . . . . 13 ⊢ (𝑛 = 2 → 𝑛 = 2)
83		1ne2 11117	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
84	83	necomi 2836	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 1
85	84	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 = 2 → 2 ≠ 1)
86	82, 85	eqnetrd 2849	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 𝑛 ≠ 1)
87	86	neneqd 2787	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 = 1)
88	87	iffalsed 4047	. . . . . . . . . 10 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
89		iftrue 4042	. . . . . . . . . 10 ⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
90	88, 89	eqtrd 2644	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
92	65	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℕ)
93	79, 91, 92, 19	fvmptd 6197	. . . . . . 7 ⊢ (𝜑 → (𝐶‘2) = 𝐵)
94	81, 93	uneq12d 3730	. . . . . 6 ⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵))
95		eqidd 2611	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵))
96	78, 94, 95	3eqtrd 2648	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
97	63, 73, 96	3eqtrd 2648	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
98	97	fveq2d 6107	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)))
99		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑛𝜑
100		nnex 10903	. . . . . . 7 ⊢ ℕ ∈ V
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
102	68	a1i 11	. . . . . 6 ⊢ (𝜑 → {1, 2} ⊆ ℕ)
103	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
104		simpl 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑)
105	102	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
106	36	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
107		elpwi 4117	. . . . . . . . 9 ⊢ ((𝐶‘𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
108	106, 107	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109	104, 105, 108	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
110	103, 109	ovncl 39457	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞))
111	58	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅))
112	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
113	112	ovn0 39456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
114	111, 113	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = 0)
115	99, 101, 102, 110, 114	sge0ss 39305	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
116	115	eqcomd 2616	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
117	81, 6	eqsstrd 3602	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ ↑𝑚 𝑋))
118	1, 117	ovncl 39457	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
119	93, 18	eqsstrd 3602	. . . . . 6 ⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ ↑𝑚 𝑋))
120	1, 119	ovncl 39457	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
121	74	fveq2d 6107	. . . . 5 ⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
122	75	fveq2d 6107	. . . . 5 ⊢ (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
123	83	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ≠ 2)
124	80, 92, 118, 120, 121, 122, 123	sge0pr 39287	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
125	81	fveq2d 6107	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
126	93	fveq2d 6107	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
127	125, 126	oveq12d 6567	. . . 4 ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
128	116, 124, 127	3eqtrd 2648	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
129	98, 128	breq12d 4596	. 2 ⊢ (𝜑 → (((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) ↔ ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))))
130	37, 129	mpbid 221	1 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {cpr 4127  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   ≤ cle 9954  ℕcn 10897  2c2 10947   +_𝑒 cxad 11820  Σ^{^}csumge0 39255  voln*covoln 39426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256  df-ovoln 39427

This theorem is referenced by:  ovnsubadd2  39536