Theorem ovnsubadd2lem 44876

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 4492	. . . . . . . 8 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
3	2	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4		ovexd 7392	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
5		ovnsubadd2lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
6	4, 5	ssexd 5281	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
7	6, 5	elpwd 4566	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
8	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
9	3, 8	eqeltrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
10	9	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
11		simpl 483	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
12	11	iffalsed 4497	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13		simpr 485	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
14	13	iftrued 4494	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
15	12, 14	eqtrd 2776	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
16	15	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
17		ovnsubadd2lem.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋))
18	4, 17	ssexd 5281	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
19	18, 17	elpwd 4566	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
20	19	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
21	16, 20	eqeltrd 2838	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
22	21	adantllr 717	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
23		simpl 483	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
24	23	iffalsed 4497	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25		simpr 485	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
26	25	iffalsed 4497	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
27	24, 26	eqtrd 2776	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28		0elpw 5311	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 (ℝ ↑m 𝑋)
29	28	a1i 11	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑m 𝑋))
30	27, 29	eqeltrd 2838	. . . . . . 7 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
31	30	adantll 712	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
32	22, 31	pm2.61dan 811	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
33	10, 32	pm2.61dan 811	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
34		ovnsubadd2lem.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
35	33, 34	fmptd 7062	. . 3 ⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ ↑m 𝑋))
36	1, 35	ovnsubadd 44803	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
37		eldifi 4086	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
38	37	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39		eldifn 4087	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40		vex 3449	. . . . . . . . . . . . . . . . 17 ⊢ 𝑛 ∈ V
41	40	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
42		id 22	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
43	41, 42	nelpr1 4614	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
44	43	neneqd 2948	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
45	39, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
46	41, 42	nelpr2 4613	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
47	46	neneqd 2948	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
48	39, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
49	45, 48, 27	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50		0ex 5264	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
52	49, 51	eqeltrd 2838	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
53	52	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
54	34	fvmpt2 6959	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
55	38, 53, 54	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
56	49	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
57	55, 56	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = ∅)
58	57	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅)
59		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑛(ℕ ∖ {1, 2})
60	59	iunxdif3 5055	. . . . . . 7 ⊢ (∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅ → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
61	58, 60	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
62	61	eqcomd 2742	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛))
63		1nn 12164	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
64		2nn 12226	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
65	63, 64	pm3.2i 471	. . . . . . . . 9 ⊢ (1 ∈ ℕ ∧ 2 ∈ ℕ)
66		prssi 4781	. . . . . . . . 9 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
67	65, 66	ax-mp 5	. . . . . . . 8 ⊢ {1, 2} ⊆ ℕ
68		dfss4 4218	. . . . . . . 8 ⊢ ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
69	67, 68	mpbi 229	. . . . . . 7 ⊢ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70		iuneq1 4970	. . . . . . 7 ⊢ ((ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2} → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛))
71	69, 70	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛)
72	71	a1i 11	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛))
73		fveq2 6842	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
74		fveq2 6842	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2))
75	73, 74	iunxprg 5056	. . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
76	63, 64, 75	mp2an 690	. . . . . . 7 ⊢ ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2))
77	76	a1i 11	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
78	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℕ)
79	34, 2, 78, 6	fvmptd3 6971	. . . . . . 7 ⊢ (𝜑 → (𝐶‘1) = 𝐴)
80		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 𝑛 = 2)
81		1ne2 12361	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
82	81	necomi 2998	. . . . . . . . . . . . 13 ⊢ 2 ≠ 1
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 2 ≠ 1)
84	80, 83	eqnetrd 3011	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → 𝑛 ≠ 1)
85	84	neneqd 2948	. . . . . . . . . 10 ⊢ (𝑛 = 2 → ¬ 𝑛 = 1)
86	85	iffalsed 4497	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
87		iftrue 4492	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
88	86, 87	eqtrd 2776	. . . . . . . 8 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
89	64	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℕ)
90	34, 88, 89, 18	fvmptd3 6971	. . . . . . 7 ⊢ (𝜑 → (𝐶‘2) = 𝐵)
91	79, 90	uneq12d 4124	. . . . . 6 ⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵))
92		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵))
93	77, 91, 92	3eqtrd 2780	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
94	62, 72, 93	3eqtrd 2780	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
95	94	fveq2d 6846	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)))
96		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑛𝜑
97		nnex 12159	. . . . . . 7 ⊢ ℕ ∈ V
98	97	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
99	67	a1i 11	. . . . . 6 ⊢ (𝜑 → {1, 2} ⊆ ℕ)
100	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
101		simpl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑)
102	99	sselda 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
103	35	ffvelcdmda 7035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
104		elpwi 4567	. . . . . . . . 9 ⊢ ((𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
105	103, 104	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
106	101, 102, 105	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
107	100, 106	ovncl 44798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞))
108	57	fveq2d 6846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅))
109	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
110	109	ovn0 44797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
111	108, 110	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = 0)
112	96, 98, 99, 107, 111	sge0ss 44643	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
113	112	eqcomd 2742	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
114	79, 5	eqsstrd 3982	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ ↑m 𝑋))
115	1, 114	ovncl 44798	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
116	90, 17	eqsstrd 3982	. . . . . 6 ⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ ↑m 𝑋))
117	1, 116	ovncl 44798	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
118		2fveq3 6847	. . . . 5 ⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
119		2fveq3 6847	. . . . 5 ⊢ (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
120	81	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ≠ 2)
121	78, 89, 115, 117, 118, 119, 120	sge0pr 44625	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
122	79	fveq2d 6846	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
123	90	fveq2d 6846	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
124	122, 123	oveq12d 7375	. . . 4 ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
125	113, 121, 124	3eqtrd 2780	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
126	95, 125	breq12d 5118	. 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))))
127	36, 126	mpbid 231	1 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ⊆ wss 3910  ∅c0 4282  ifcif 4486  𝒫 cpw 4560  {cpr 4588  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Fincfn 8883  ℝcr 11050  0cc0 11051  1c1 11052   ≤ cle 11190  ℕcn 12153  2c2 12208   +_𝑒 cxad 13031  Σ^{^}csumge0 44593  voln*covoln 44767

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-0g 17323  df-topgen 17325  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594  df-ovoln 44768

This theorem is referenced by:  ovnsubadd2  44877