Theorem ovnsubadd2lem 46660

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 4531	. . . . . . . 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		ovexd 7466	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
5		ovnsubadd2lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
6	4, 5	ssexd 5324	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
7	6, 5	elpwd 4606	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
9	3, 8	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
10	9	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
11		simpl 482	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
12	11	iffalsed 4536	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13		simpr 484	. . . . . . . . . . 11 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
14	13	iftrued 4533	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
15	12, 14	eqtrd 2777	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
16	15	adantll 714	. . . . . . . 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 5324	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
19	18, 17	elpwd 4606	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
20	19	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
21	16, 20	eqeltrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
22	21	adantllr 719	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
23		simpl 482	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
24	23	iffalsed 4536	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25		simpr 484	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
26	25	iffalsed 4536	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
27	24, 26	eqtrd 2777	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28		0elpw 5356	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 (ℝ ↑m 𝑋)
29	28	a1i 11	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑m 𝑋))
30	27, 29	eqeltrd 2841	. . . . . . 7 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
31	30	adantll 714	. . . . . 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 813	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
33	10, 32	pm2.61dan 813	. . . 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 7134	. . 3 ⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ ↑m 𝑋))
36	1, 35	ovnsubadd 46587	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
37		eldifi 4131	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
38	37	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39		eldifn 4132	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40		vex 3484	. . . . . . . . . . . . . . . . 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 4654	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
44	43	neneqd 2945	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
45	39, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
46	41, 42	nelpr2 4653	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
47	46	neneqd 2945	. . . . . . . . . . . . . 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 5307	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
52	49, 51	eqeltrd 2841	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
53	52	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
54	34	fvmpt2 7027	. . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
57	55, 56	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = ∅)
58	57	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅)
59		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑛(ℕ ∖ {1, 2})
60	59	iunxdif3 5095	. . . . . . 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 2743	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛))
63		1nn 12277	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
64		2nn 12339	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
65	63, 64	pm3.2i 470	. . . . . . . . 9 ⊢ (1 ∈ ℕ ∧ 2 ∈ ℕ)
66		prssi 4821	. . . . . . . . 9 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
67	65, 66	ax-mp 5	. . . . . . . 8 ⊢ {1, 2} ⊆ ℕ
68		dfss4 4269	. . . . . . . 8 ⊢ ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
69	67, 68	mpbi 230	. . . . . . 7 ⊢ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70		iuneq1 5008	. . . . . . 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 6906	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
74		fveq2 6906	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2))
75	73, 74	iunxprg 5096	. . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
76	63, 64, 75	mp2an 692	. . . . . . 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 7039	. . . . . . 7 ⊢ (𝜑 → (𝐶‘1) = 𝐴)
80		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 𝑛 = 2)
81		1ne2 12474	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
82	81	necomi 2995	. . . . . . . . . . . . 13 ⊢ 2 ≠ 1
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 2 ≠ 1)
84	80, 83	eqnetrd 3008	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → 𝑛 ≠ 1)
85	84	neneqd 2945	. . . . . . . . . 10 ⊢ (𝑛 = 2 → ¬ 𝑛 = 1)
86	85	iffalsed 4536	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
87		iftrue 4531	. . . . . . . . 9 ⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
88	86, 87	eqtrd 2777	. . . . . . . 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 7039	. . . . . . 7 ⊢ (𝜑 → (𝐶‘2) = 𝐵)
91	79, 90	uneq12d 4169	. . . . . 6 ⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵))
92		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵))
93	77, 91, 92	3eqtrd 2781	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
94	62, 72, 93	3eqtrd 2781	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
95	94	fveq2d 6910	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)))
96		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑛𝜑
97		nnex 12272	. . . . . . 7 ⊢ ℕ ∈ V
98	97	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
99	67	a1i 11	. . . . . 6 ⊢ (𝜑 → {1, 2} ⊆ ℕ)
100	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
101		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑)
102	99	sselda 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
103	35	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
104		elpwi 4607	. . . . . . . . 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 46582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞))
108	57	fveq2d 6910	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅))
109	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
110	109	ovn0 46581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
111	108, 110	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = 0)
112	96, 98, 99, 107, 111	sge0ss 46427	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
113	112	eqcomd 2743	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
114	79, 5	eqsstrd 4018	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ ↑m 𝑋))
115	1, 114	ovncl 46582	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
116	90, 17	eqsstrd 4018	. . . . . 6 ⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ ↑m 𝑋))
117	1, 116	ovncl 46582	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
118		2fveq3 6911	. . . . 5 ⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
119		2fveq3 6911	. . . . 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 46409	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
122	79	fveq2d 6910	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
123	90	fveq2d 6910	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
124	122, 123	oveq12d 7449	. . . 4 ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
125	113, 121, 124	3eqtrd 2781	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
126	95, 125	breq12d 5156	. 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 232	1 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  ifcif 4525  𝒫 cpw 4600  {cpr 4628  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  Fincfn 8985  ℝcr 11154  0cc0 11155  1c1 11156   ≤ cle 11296  ℕcn 12266  2c2 12321   +_𝑒 cxad 13152  Σ^{^}csumge0 46377  voln*covoln 46551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-prod 15940  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-sumge0 46378  df-ovoln 46552

This theorem is referenced by:  ovnsubadd2  46661