Theorem volsupnfl 33787

Description: volsup 23544 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)

Hypothesis

Ref	Expression
volsupnfl.0	⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))

Assertion

Ref	Expression
volsupnfl	⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Distinct variable group:   𝑓,𝑛,𝑥,𝐴

Proof of Theorem volsupnfl

Dummy variables 𝑔 𝑚 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4582	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4601	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	syl6eq 2821	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6336	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol‘∪ 𝐴) = (vol‘∅))
5		0mbl 23527	. . . . . . . . 9 ⊢ ∅ ∈ dom vol
6		mblvol 23518	. . . . . . . . 9 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (vol‘∅) = (vol*‘∅)
8		ovol0 23481	. . . . . . . 8 ⊢ (vol*‘∅) = 0
9	7, 8	eqtri 2793	. . . . . . 7 ⊢ (vol‘∅) = 0
10	4, 9	syl6req 2822	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol‘∪ 𝐴))
11	10	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴)))
12		reldom 8115	. . . . . . . . . . 11 ⊢ Rel ≼
13	12	brrelexi 5298	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
14		0sdomg 8245	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
16	15	biimparc 465	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17		fodomr 8267	. . . . . . . 8 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→𝐴)
18	16, 17	sylancom 576	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→𝐴)
19		unissb 4605	. . . . . . . . . . . . 13 ⊢ (∪ 𝐴 ⊆ ℝ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ)
20	19	anbi1i 610	. . . . . . . . . . . 12 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
21		r19.26 3212	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
22	20, 21	bitr4i 267	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23		ovolctb2 23480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24		nulmbl 23523	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25		mblvol 23518	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26		eqtr 2790	. . . . . . . . . . . . . . . . 17 ⊢ (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
27	26	expcom 398	. . . . . . . . . . . . . . . 16 ⊢ ((vol*‘𝑥) = 0 → ((vol‘𝑥) = (vol*‘𝑥) → (vol‘𝑥) = 0))
28	25, 27	syl5 34	. . . . . . . . . . . . . . 15 ⊢ ((vol*‘𝑥) = 0 → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
29	28	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
30	24, 29	jcai 506	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
31	23, 30	syldan 579	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
32	31	ralimi 3101	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
33	22, 32	sylbi 207	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
34	33	ancoms 455	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35		fzfi 12979	. . . . . . . . . . . . . . 15 ⊢ (1...𝑚) ∈ Fin
36		fzssuz 12589	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑚) ⊆ (ℤ≥‘1)
37		nnuz 11925	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ≥‘1)
38	36, 37	sseqtr4i 3787	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ⊆ ℕ
39		fof 6256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔:ℕ⟶𝐴)
40	39	ffvelrnda 6502	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) → (𝑔‘𝑙) ∈ 𝐴)
41		eleq1 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑔‘𝑙) → (𝑥 ∈ dom vol ↔ (𝑔‘𝑙) ∈ dom vol))
42		fveq2 6332	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑔‘𝑙) → (vol‘𝑥) = (vol‘(𝑔‘𝑙)))
43	42	eqeq1d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑔‘𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔‘𝑙)) = 0))
44	41, 43	anbi12d 616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑔‘𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔‘𝑙) ∈ dom vol ∧ (vol‘(𝑔‘𝑙)) = 0)))
45	44	rspccva 3459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → ((𝑔‘𝑙) ∈ dom vol ∧ (vol‘(𝑔‘𝑙)) = 0))
46	45	simpld 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → (𝑔‘𝑙) ∈ dom vol)
47	46	ancoms 455	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔‘𝑙) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔‘𝑙) ∈ dom vol)
48	40, 47	sylan 569	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔‘𝑙) ∈ dom vol)
49	48	an32s 631	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔‘𝑙) ∈ dom vol)
50	49	ralrimiva 3115	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔‘𝑙) ∈ dom vol)
51		ssralv 3815	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (𝑔‘𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol))
52	38, 50, 51	mpsyl 68	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
53		finiunmbl 23532	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
54	35, 52, 53	sylancr 575	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
55	54	adantr 466	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
56		eqid 2771	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
57	55, 56	fmptd 6527	. . . . . . . . . . . 12 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol)
58		fzssp1 12591	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ⊆ (1...(𝑛 + 1))
59		iunss1 4666	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑛) ⊆ (1...(𝑛 + 1)) → ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ⊆ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
60	58, 59	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ⊆ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙)
61		oveq2 6801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
62	61	iuneq1d 4679	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙))
63		ovex 6823	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ∈ V
64		fvex 6342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔‘𝑙) ∈ V
65	63, 64	iunex 7294	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ∈ V
66	62, 56, 65	fvmpt 6424	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) = ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙))
67		peano2nn 11234	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
68		oveq2 6801	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
69	68	iuneq1d 4679	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 + 1) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
70		ovex 6823	. . . . . . . . . . . . . . . . . 18 ⊢ (1...(𝑛 + 1)) ∈ V
71	70, 64	iunex 7294	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙) ∈ V
72	69, 56, 71	fvmpt 6424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
73	67, 72	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
74	66, 73	sseq12d 3783	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)) ↔ ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ⊆ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙)))
75	60, 74	mpbiri 248	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)))
76	75	rgen 3071	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))
77		nnex 11228	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
78	77	mptex 6630	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ V
79		feq1 6166	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol))
80		fveq1 6331	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓‘𝑛) = ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛))
81		fveq1 6331	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)))
82	80, 81	sseq12d 3783	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ((𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1)) ↔ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))))
83	82	ralbidv 3135	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1)) ↔ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))))
84	79, 83	anbi12d 616	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) ↔ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)))))
85		rneq 5489	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
86	85	unieqd 4584	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ∪ ran 𝑓 = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
87	86	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (vol‘∪ ran 𝑓) = (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))))
88	85	imaeq2d 5607	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))))
89	88	supeq1d 8508	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → sup((vol “ ran 𝑓), ℝ*, < ) = sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ))
90	87, 89	eqeq12d 2786	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ((vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ) ↔ (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < )))
91	84, 90	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < )) ↔ (((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ))))
92		volsupnfl.0	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))
93	78, 91, 92	vtocl 3410	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ))
94	57, 76, 93	sylancl 574	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ))
95		df-iun 4656	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥)}
96		eluzfz2 12556	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (ℤ≥‘1) → 𝑥 ∈ (1...𝑥))
97	96, 37	eleq2s 2868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
98		fveq2 6332	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑥 → (𝑔‘𝑙) = (𝑔‘𝑥))
99	98	eleq2d 2836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑥 → (𝑛 ∈ (𝑔‘𝑙) ↔ 𝑛 ∈ (𝑔‘𝑥)))
100	99	rspcev 3460	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙))
101	97, 100	sylan 569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙))
102		oveq2 6801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
103	102	rexeqdv 3294	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙)))
104	103	rspcev 3460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
105	101, 104	syldan 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
106	105	rexlimiva 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
107		ssrexv 3816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1...𝑚) ⊆ ℕ → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙)))
108	38, 107	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙))
109	99	cbvrexv 3321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙) ↔ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
110	108, 109	sylib 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
111	110	rexlimivw 3177	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
112	106, 111	impbii 199	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
113		eliun 4658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
114	113	rexbii 3189	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
115	112, 114	bitr4i 267	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
116	115	abbii 2888	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
117	95, 116	eqtri 2793	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
118		df-iun 4656	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑚 ∈ ℕ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
119		ovex 6823	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑚) ∈ V
120	119, 64	iunex 7294	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ V
121	120	dfiun3 5518	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑚 ∈ ℕ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
122	117, 118, 121	3eqtr2i 2799	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
123		fofn 6258	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
124		fniunfv 6648	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 Fn ℕ → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran 𝑔)
125	123, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran 𝑔)
126		forn 6259	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ–onto→𝐴 → ran 𝑔 = 𝐴)
127	126	unieqd 4584	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ ran 𝑔 = ∪ 𝐴)
128	125, 127	eqtrd 2805	. . . . . . . . . . . . . 14 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ 𝐴)
129	122, 128	syl5eqr 2819	. . . . . . . . . . . . 13 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = ∪ 𝐴)
130	129	fveq2d 6336	. . . . . . . . . . . 12 ⊢ (𝑔:ℕ–onto→𝐴 → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol‘∪ 𝐴))
131	130	adantr 466	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol‘∪ 𝐴))
132		rnco2 5786	. . . . . . . . . . . . . 14 ⊢ ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
133		eqidd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
134		volf 23517	. . . . . . . . . . . . . . . . . . 19 ⊢ vol:dom vol⟶(0[,]+∞)
135	134	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
136	135	feqmptd 6391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
137		fveq2 6332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) → (vol‘𝑛) = (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
138	55, 133, 136, 137	fmptco 6539	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (𝑚 ∈ ℕ ↦ (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))))
139		mblvol 23518	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol → (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
140	55, 139	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
141		mblss 23519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
142	141	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
143	25	eqeq1d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
144		0re 10242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℝ
145		eleq1a 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
146	144, 145	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
147	143, 146	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
148	147	imp 393	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
149	142, 148	jca 501	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150	149	ralimi 3101	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151	150	adantl 467	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
152		ssid 3773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℕ ⊆ ℕ
153		sseq1 3775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑔‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔‘𝑙) ⊆ ℝ))
154		fveq2 6332	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑔‘𝑙) → (vol*‘𝑥) = (vol*‘(𝑔‘𝑙)))
155	154	eleq1d 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑔‘𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
156	153, 155	anbi12d 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝑔‘𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
157	156	ralima 6641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
158	123, 152, 157	sylancl 574	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
159		foima 6261	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑔:ℕ–onto→𝐴 → (𝑔 “ ℕ) = 𝐴)
160	159	raleqdv 3293	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161	158, 160	bitr3d 270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
162	161	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
163	151, 162	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
164		ssralv 3815	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ) → ∀𝑙 ∈ (1...𝑚)((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
165	38, 163, 164	mpsyl 68	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
166	165	adantr 466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
167		ovolfiniun 23489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)))
168	35, 166, 167	sylancr 575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)))
169		mblvol 23518	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔‘𝑙) ∈ dom vol → (vol‘(𝑔‘𝑙)) = (vol*‘(𝑔‘𝑙)))
170	49, 169	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔‘𝑙)) = (vol*‘(𝑔‘𝑙)))
171	45	simprd 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → (vol‘(𝑔‘𝑙)) = 0)
172	40, 171	sylan2 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ)) → (vol‘(𝑔‘𝑙)) = 0)
173	172	ancoms 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔‘𝑙)) = 0)
174	173	an32s 631	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔‘𝑙)) = 0)
175	170, 174	eqtr3d 2807	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔‘𝑙)) = 0)
176	175	ralrimiva 3115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔‘𝑙)) = 0)
177		ssralv 3815	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (vol*‘(𝑔‘𝑙)) = 0 → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0))
178	38, 176, 177	mpsyl 68	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
179	178	adantr 466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
180	179	sumeq2d 14640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = Σ𝑙 ∈ (1...𝑚)0)
181	35	olci 855	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑚) ⊆ (ℤ≥‘1) ∨ (1...𝑚) ∈ Fin)
182		sumz 14661	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑚) ⊆ (ℤ≥‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑙 ∈ (1...𝑚)0 = 0)
183	181, 182	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑙 ∈ (1...𝑚)0 = 0
184	180, 183	syl6eq 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
185	168, 184	breqtrd 4812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ 0)
186		mblss 23519	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘𝑙) ∈ dom vol → (𝑔‘𝑙) ⊆ ℝ)
187	186	ralimi 3101	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
188	52, 187	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
189		iunss 4695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ ↔ ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
190	188, 189	sylibr 224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
191	190	adantr 466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
192		ovolge0 23469	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ → 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
193	191, 192	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
194		ovolcl 23466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
195	190, 194	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
196	195	adantr 466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
197		0xr 10288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ*
198		xrletri3 12190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0 ↔ ((vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))))
199	196, 197, 198	sylancl 574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ((vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0 ↔ ((vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))))
200	185, 193, 199	mpbir2and 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0)
201	140, 200	eqtrd 2805	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0)
202	201	mpteq2dva 4878	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (𝑚 ∈ ℕ ↦ 0))
203		fconstmpt 5303	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
204	202, 203	syl6eqr 2823	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}))
205	138, 204	eqtrd 2805	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}))
206		frn 6193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ⊆ dom vol)
207		ffn 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
208	134, 207	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ vol Fn dom vol
209	120, 56	fnmpti 6162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) Fn ℕ
210		fnco 6139	. . . . . . . . . . . . . . . . . 18 ⊢ ((vol Fn dom vol ∧ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) Fn ℕ ∧ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ⊆ dom vol) → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) Fn ℕ)
211	208, 209, 210	mp3an12 1562	. . . . . . . . . . . . . . . . 17 ⊢ (ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ⊆ dom vol → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) Fn ℕ)
212	57, 206, 211	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) Fn ℕ)
213		1nn 11233	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ
214	213	ne0ii 4071	. . . . . . . . . . . . . . . 16 ⊢ ℕ ≠ ∅
215		fconst5 6615	. . . . . . . . . . . . . . . 16 ⊢ (((vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) Fn ℕ ∧ ℕ ≠ ∅) → ((vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}) ↔ ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0}))
216	212, 214, 215	sylancl 574	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ((vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}) ↔ ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0}))
217	205, 216	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0})
218	132, 217	syl5eqr 2819	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0})
219	218	supeq1d 8508	. . . . . . . . . . . 12 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ) = sup({0}, ℝ*, < ))
220		xrltso 12179	. . . . . . . . . . . . 13 ⊢ < Or ℝ*
221		supsn 8534	. . . . . . . . . . . . 13 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
222	220, 197, 221	mp2an 672	. . . . . . . . . . . 12 ⊢ sup({0}, ℝ*, < ) = 0
223	219, 222	syl6eq 2821	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))), ℝ*, < ) = 0)
224	94, 131, 223	3eqtr3rd 2814	. . . . . . . . . 10 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘∪ 𝐴))
225	224	ex 397	. . . . . . . . 9 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 0 = (vol‘∪ 𝐴)))
226	34, 225	syl5 34	. . . . . . . 8 ⊢ (𝑔:ℕ–onto→𝐴 → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
227	226	exlimiv 2010	. . . . . . 7 ⊢ (∃𝑔 𝑔:ℕ–onto→𝐴 → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
228	18, 227	syl 17	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
229	228	expimpd 441	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴)))
230	11, 229	pm2.61ine 3026	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴))
231		renepnf 10289	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
232	144, 231	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
233		fveq2 6332	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol‘∪ 𝐴) = (vol‘ℝ))
234		rembl 23528	. . . . . . . . 9 ⊢ ℝ ∈ dom vol
235		mblvol 23518	. . . . . . . . 9 ⊢ (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
236	234, 235	ax-mp 5	. . . . . . . 8 ⊢ (vol‘ℝ) = (vol*‘ℝ)
237		ovolre 23513	. . . . . . . 8 ⊢ (vol*‘ℝ) = +∞
238	236, 237	eqtri 2793	. . . . . . 7 ⊢ (vol‘ℝ) = +∞
239	233, 238	syl6eq 2821	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol‘∪ 𝐴) = +∞)
240	232, 239	neeqtrrd 3017	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol‘∪ 𝐴))
241	240	necon2i 2977	. . . 4 ⊢ (0 = (vol‘∪ 𝐴) → ∪ 𝐴 ≠ ℝ)
242	230, 241	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → ∪ 𝐴 ≠ ℝ)
243	242	expr 444	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
244		eqimss 3806	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
245	244	necon3bi 2969	. 2 ⊢ (¬ ∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ)
246	243, 245	pm2.61d1 172	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 836   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ⊆ wss 3723  ∅c0 4063  {csn 4316  ∪ cuni 4574  ∪ ciun 4654   class class class wbr 4786   ↦ cmpt 4863   Or wor 5169   × cxp 5247  dom cdm 5249  ran crn 5250   “ cima 5252   ∘ ccom 5253   Fn wfn 6026  ⟶wf 6027  –onto→wfo 6029  ‘cfv 6031  (class class class)co 6793   ≼ cdom 8107   ≺ csdm 8108  Fincfn 8109  supcsup 8502  ℝcr 10137  0cc0 10138  1c1 10139   + caddc 10141  +∞cpnf 10273  ℝ^*cxr 10275   < clt 10276   ≤ cle 10277  ℕcn 11222  ℤ_≥cuz 11888  [,]cicc 12383  ...cfz 12533  Σcsu 14624  vol*covol 23450  volcvol 23451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cmp 21411  df-ovol 23452  df-vol 23453

This theorem is referenced by: (None)