Theorem ovnsubadd2lem 47791

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 4487	. . . . . . . 8 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
3	2	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4		ovexd 7459	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
5		ovnsubadd2lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
6	4, 5	ssexd 5286	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
7	6, 5	elpwd 4562	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
8	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
9	3, 8	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
10	9	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
11		simpl 488	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
12	11	iffalsed 4492	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13		simpr 490	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
14	13	iftrued 4489	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
15	12, 14	eqtrd 2795	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
16	15	adantll 727	. . . . . . . 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 5286	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
19	18, 17	elpwd 4562	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
20	19	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
21	16, 20	eqeltrd 2860	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
22	21	adantllr 732	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
23		simpl 488	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
24	23	iffalsed 4492	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25		simpr 490	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
26	25	iffalsed 4492	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
27	24, 26	eqtrd 2795	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28		0elpw 5317	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 (ℝ ↑m 𝑋)
29	28	a1i 11	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑m 𝑋))
30	27, 29	eqeltrd 2860	. . . . . . 7 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
31	30	adantll 727	. . . . . 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 825	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
33	10, 32	pm2.61dan 825	. . . 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 7119	. . 3 ⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ ↑m 𝑋))
36	1, 35	ovnsubadd 47718	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
37		eldifi 4066	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
38	37	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39		eldifn 4067	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑛 ∈ V
41	40	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
42		id 23	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
43	41, 42	nelpr1 4611	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
44	43	neneqd 2960	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
45	39, 44	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
46	41, 42	nelpr2 4610	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
47	46	neneqd 2960	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
48	39, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
49	45, 48, 27	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50		0ex 5260	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
52	49, 51	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
53	52	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
54	34	fvmpt2 7010	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
56	49	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
57	55, 56	eqtrd 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = ∅)
58	57	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅)
59		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑛(ℕ ∖ {1, 2})
60	59	iunxdif3 5054	. . . . . . 7 ⊢ (∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅ → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
61	58, 60	syl 18	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛))
62	61	eqcomd 2766	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛))
63		1nn 12353	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
64		2nn 12423	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
65	63, 64	pm3.2i 476	. . . . . . . . 9 ⊢ (1 ∈ ℕ ∧ 2 ∈ ℕ)
66		prssi 4776	. . . . . . . . 9 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
67	65, 66	ax-mp 5	. . . . . . . 8 ⊢ {1, 2} ⊆ ℕ
68		dfss4 4213	. . . . . . . 8 ⊢ ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
69	67, 68	mpbi 233	. . . . . . 7 ⊢ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70		iuneq1 4967	. . . . . . 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 6890	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
74		fveq2 6890	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2))
75	73, 74	iunxprg 5055	. . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
76	63, 64, 75	mp2an 705	. . . . . . 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 7022	. . . . . . 7 ⊢ (𝜑 → (𝐶‘1) = 𝐴)
80		id 23	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 𝑛 = 2)
81		1ne2 12560	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
82	81	necomi 3009	. . . . . . . . . . . . 13 ⊢ 2 ≠ 1
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 2 ≠ 1)
84	80, 83	eqnetrd 3022	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → 𝑛 ≠ 1)
85	84	neneqd 2960	. . . . . . . . . 10 ⊢ (𝑛 = 2 → ¬ 𝑛 = 1)
86	85	iffalsed 4492	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
87		iftrue 4487	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
88	86, 87	eqtrd 2795	. . . . . . . 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 7022	. . . . . . 7 ⊢ (𝜑 → (𝐶‘2) = 𝐵)
91	79, 90	uneq12d 4167	. . . . . 6 ⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵))
92		eqidd 2761	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵))
93	77, 91, 92	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
94	62, 72, 93	3eqtrd 2799	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
95	94	fveq2d 6894	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)))
96		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑛𝜑
97		nnex 12348	. . . . . . 7 ⊢ ℕ ∈ V
98	97	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
99	67	a1i 11	. . . . . 6 ⊢ (𝜑 → {1, 2} ⊆ ℕ)
100	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
101		simpl 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑)
102	99	sselda 3909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
103	35	ffvelcdmda 7089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
104		elpwi 4563	. . . . . . . . 9 ⊢ ((𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
105	103, 104	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
106	101, 102, 105	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
107	100, 106	ovncl 47713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞))
108	57	fveq2d 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅))
109	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
110	109	ovn0 47712	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
111	108, 110	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = 0)
112	96, 98, 99, 107, 111	sge0ss 47558	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
113	112	eqcomd 2766	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
114	79, 5	eqsstrd 3943	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ ↑m 𝑋))
115	1, 114	ovncl 47713	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
116	90, 17	eqsstrd 3943	. . . . . 6 ⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ ↑m 𝑋))
117	1, 116	ovncl 47713	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
118		2fveq3 6895	. . . . 5 ⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
119		2fveq3 6895	. . . . 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 47540	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +e ((voln*‘𝑋)‘(𝐶‘2))))
122	79	fveq2d 6894	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
123	90	fveq2d 6894	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
124	122, 123	oveq12d 7442	. . . 4 ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +e ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +e ((voln*‘𝑋)‘𝐵)))
125	113, 121, 124	3eqtrd 2799	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +e ((voln*‘𝑋)‘𝐵)))
126	95, 125	breq12d 5115	. 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 235	1 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +e ((voln*‘𝑋)‘𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   ∪ cun 4148  ∅c0 4277  ifcif 4481  𝒫 cpw 4556  {cpr 4586  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424   ↑_m cmap 8855  Fincfn 8981  ℝcr 11206  0cc0 11207  1c1 11208   ≤ cle 11351  ℕcn 12342  2c2 12404   +_e cxad 13246  Σ^{^}csumge0 47508  voln*covoln 47682

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cc 10520  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-er 8725  df-map 8857  df-pm 8858  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fi 9410  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-xneg 13248  df-xadd 13249  df-xmul 13250  df-ioo 13487  df-ico 13489  df-icc 13490  df-fz 13647  df-fzo 13796  df-fl 13939  df-seq 14152  df-exp 14212  df-hash 14482  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-rlim 15665  df-sum 15863  df-prod 16082  df-rest 17602  df-topgen 17623  df-psmet 21679  df-xmet 21680  df-met 21681  df-bl 21682  df-mopn 21683  df-top 23221  df-topon 23238  df-bases 23273  df-cmp 23714  df-ovol 25794  df-vol 25795  df-sumge0 47509  df-ovoln 47683

This theorem is used by:  ovnsubadd2  47792