Theorem ovnsubadd2lem 46092

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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4		ovexd 7448	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
5		ovnsubadd2lem.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
6	4, 5	ssexd 5320	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
7	6, 5	elpwd 4605	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
8	7	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
9	3, 8	eqeltrd 2825	. . . . . 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 481	. . . . . . . . . . 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 483	. . . . . . . . . . 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 2765	. . . . . . . . 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 5320	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
19	18, 17	elpwd 4605	. . . . . . . . 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 2825	. . . . . . 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 481	. . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
26	25	iffalsed 4536	. . . . . . . . 9 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
27	24, 26	eqtrd 2765	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28		0elpw 5351	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 (ℝ ↑m 𝑋)
29	28	a1i 11	. . . . . . . 8 ⊢ ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑m 𝑋))
30	27, 29	eqeltrd 2825	. . . . . . 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 7117	. . 3 ⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ ↑m 𝑋))
36	1, 35	ovnsubadd 46019	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
37		eldifi 4120	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
38	37	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39		eldifn 4121	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40		vex 3467	. . . . . . . . . . . . . . . . 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 4653	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
44	43	neneqd 2935	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
45	39, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
46	41, 42	nelpr2 4652	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
47	46	neneqd 2935	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
48	39, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
49	45, 48, 27	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50		0ex 5303	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
52	49, 51	eqeltrd 2825	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
53	52	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
54	34	fvmpt2 7009	. . . . . . . . . 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 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
56	49	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
57	55, 56	eqtrd 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶‘𝑛) = ∅)
58	57	ralrimiva 3136	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅)
59		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑛(ℕ ∖ {1, 2})
60	59	iunxdif3 5094	. . . . . . 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 2731	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶‘𝑛))
63		1nn 12248	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
64		2nn 12310	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
65	63, 64	pm3.2i 469	. . . . . . . . 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 4254	. . . . . . . 8 ⊢ ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
69	67, 68	mpbi 229	. . . . . . 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 6890	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
74		fveq2 6890	. . . . . . . . 9 ⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2))
75	73, 74	iunxprg 5095	. . . . . . . 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 7021	. . . . . . 7 ⊢ (𝜑 → (𝐶‘1) = 𝐴)
80		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 𝑛 = 2)
81		1ne2 12445	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
82	81	necomi 2985	. . . . . . . . . . . . 13 ⊢ 2 ≠ 1
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → 2 ≠ 1)
84	80, 83	eqnetrd 2998	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → 𝑛 ≠ 1)
85	84	neneqd 2935	. . . . . . . . . 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 2765	. . . . . . . 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 7021	. . . . . . 7 ⊢ (𝜑 → (𝐶‘2) = 𝐵)
91	79, 90	uneq12d 4158	. . . . . 6 ⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵))
92		eqidd 2726	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵))
93	77, 91, 92	3eqtrd 2769	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵))
94	62, 72, 93	3eqtrd 2769	. . . 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 1909	. . . . . 6 ⊢ Ⅎ𝑛𝜑
97		nnex 12243	. . . . . . 7 ⊢ ℕ ∈ V
98	97	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
99	67	a1i 11	. . . . . 6 ⊢ (𝜑 → {1, 2} ⊆ ℕ)
100	1	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
101		simpl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑)
102	99	sselda 3973	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
103	35	ffvelcdmda 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
104		elpwi 4606	. . . . . . . . 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 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋))
107	100, 106	ovncl 46014	. . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
110	109	ovn0 46013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
111	108, 110	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶‘𝑛)) = 0)
112	96, 98, 99, 107, 111	sge0ss 45859	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
113	112	eqcomd 2731	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))))
114	79, 5	eqsstrd 4012	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ ↑m 𝑋))
115	1, 114	ovncl 46014	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
116	90, 17	eqsstrd 4012	. . . . . 6 ⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ ↑m 𝑋))
117	1, 116	ovncl 46014	. . . . 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 45841	. . . 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 7431	. . . 4 ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
125	113, 121, 124	3eqtrd 2769	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
126	95, 125	breq12d 5157	. 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 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  Vcvv 3463   ∖ cdif 3938   ∪ cun 3939   ⊆ wss 3941  ∅c0 4319  ifcif 4525  𝒫 cpw 4599  {cpr 4627  ∪ ciun 4992   class class class wbr 5144   ↦ cmpt 5227  ‘cfv 6543  (class class class)co 7413   ↑_m cmap 8838  Fincfn 8957  ℝcr 11132  0cc0 11133  1c1 11134   ≤ cle 11274  ℕcn 12237  2c2 12292   +_𝑒 cxad 13117  Σ^{^}csumge0 45809  voln*covoln 45983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cc 10453  ax-ac2 10481  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211  ax-addf 11212  ax-mulf 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-disj 5110  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9429  df-sup 9460  df-inf 9461  df-oi 9528  df-dju 9919  df-card 9957  df-acn 9960  df-ac 10134  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-q 12958  df-rp 13002  df-xneg 13119  df-xadd 13120  df-xmul 13121  df-ioo 13355  df-ico 13357  df-icc 13358  df-fz 13512  df-fzo 13655  df-fl 13784  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-rlim 15460  df-sum 15660  df-prod 15877  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-rest 17398  df-0g 17417  df-topgen 17419  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18892  df-minusg 18893  df-subg 19077  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20272  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-drng 20625  df-psmet 21270  df-xmet 21271  df-met 21272  df-bl 21273  df-mopn 21274  df-cnfld 21279  df-top 22809  df-topon 22826  df-bases 22862  df-cmp 23304  df-ovol 25406  df-vol 25407  df-sumge0 45810  df-ovoln 45984

This theorem is referenced by:  ovnsubadd2  46093