Theorem volsupnfl 33761

Description: volsup 23531 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 4631	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4652	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	syl6eq 2852	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6406	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol‘∪ 𝐴) = (vol‘∅))
5		0mbl 23514	. . . . . . . . 9 ⊢ ∅ ∈ dom vol
6		mblvol 23505	. . . . . . . . 9 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (vol‘∅) = (vol*‘∅)
8		ovol0 23468	. . . . . . . 8 ⊢ (vol*‘∅) = 0
9	7, 8	eqtri 2824	. . . . . . 7 ⊢ (vol‘∅) = 0
10	4, 9	syl6req 2853	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol‘∪ 𝐴))
11	10	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴)))
12		reldom 8192	. . . . . . . . . . 11 ⊢ Rel ≼
13	12	brrelexi 5352	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
14		0sdomg 8322	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
16	15	biimparc 467	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17		fodomr 8344	. . . . . . . 8 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→𝐴)
18	16, 17	sylancom 578	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→𝐴)
19		unissb 4656	. . . . . . . . . . . . 13 ⊢ (∪ 𝐴 ⊆ ℝ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ)
20	19	anbi1i 612	. . . . . . . . . . . 12 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
21		r19.26 3248	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
22	20, 21	bitr4i 269	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23		ovolctb2 23467	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24		nulmbl 23510	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25		mblvol 23505	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26		eqtr 2821	. . . . . . . . . . . . . . . . 17 ⊢ (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
27	26	expcom 400	. . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
30	24, 29	jcai 508	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
31	23, 30	syldan 581	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
32	31	ralimi 3136	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
33	22, 32	sylbi 208	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
34	33	ancoms 448	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35		fzfi 12989	. . . . . . . . . . . . . . 15 ⊢ (1...𝑚) ∈ Fin
36		fzssuz 12599	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑚) ⊆ (ℤ≥‘1)
37		nnuz 11935	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ≥‘1)
38	36, 37	sseqtr4i 3829	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ⊆ ℕ
39		fof 6325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔:ℕ⟶𝐴)
40	39	ffvelrnda 6575	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) → (𝑔‘𝑙) ∈ 𝐴)
41		eleq1 2869	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑔‘𝑙) → (𝑥 ∈ dom vol ↔ (𝑔‘𝑙) ∈ dom vol))
42		fveqeq2 6411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑔‘𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔‘𝑙)) = 0))
43	41, 42	anbi12d 618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑔‘𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔‘𝑙) ∈ dom vol ∧ (vol‘(𝑔‘𝑙)) = 0)))
44	43	rspccva 3497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → ((𝑔‘𝑙) ∈ dom vol ∧ (vol‘(𝑔‘𝑙)) = 0))
45	44	simpld 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → (𝑔‘𝑙) ∈ dom vol)
46	45	ancoms 448	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔‘𝑙) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔‘𝑙) ∈ dom vol)
47	40, 46	sylan 571	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔‘𝑙) ∈ dom vol)
48	47	an32s 634	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔‘𝑙) ∈ dom vol)
49	48	ralrimiva 3150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔‘𝑙) ∈ dom vol)
50		ssralv 3857	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (𝑔‘𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol))
51	38, 49, 50	mpsyl 68	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
52		finiunmbl 23519	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
53	35, 51, 52	sylancr 577	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
54	53	adantr 468	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol)
55	54	fmpttd 6601	. . . . . . . . . . . 12 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol)
56		fzssp1 12601	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ⊆ (1...(𝑛 + 1))
57		iunss1 4717	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑛) ⊆ (1...(𝑛 + 1)) → ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ⊆ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
58	56, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ⊆ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙)
59		oveq2 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
60	59	iuneq1d 4730	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙))
61		eqid 2802	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
62		ovex 6900	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ∈ V
63		fvex 6415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔‘𝑙) ∈ V
64	62, 63	iunex 7371	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙) ∈ V
65	60, 61, 64	fvmpt 6497	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) = ∪ 𝑙 ∈ (1...𝑛)(𝑔‘𝑙))
66		peano2nn 11311	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
67		oveq2 6876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
68	67	iuneq1d 4730	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 + 1) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
69		ovex 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (1...(𝑛 + 1)) ∈ V
70	69, 63	iunex 7371	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙) ∈ V
71	68, 61, 70	fvmpt 6497	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
72	66, 71	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)) = ∪ 𝑙 ∈ (1...(𝑛 + 1))(𝑔‘𝑙))
73	65, 72	sseq12d 3825	. . . . . . . . . . . . . 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)))
74	58, 73	mpbiri 249	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)))
75	74	rgen 3106	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1))
76		nnex 11305	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
77	76	mptex 6705	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ V
78		feq1 6231	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol))
79		fveq1 6401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓‘𝑛) = ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘𝑛))
80		fveq1 6401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))‘(𝑛 + 1)))
81	79, 80	sseq12d 3825	. . . . . . . . . . . . . . . 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))))
82	81	ralbidv 3170	. . . . . . . . . . . . . . 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))))
83	78, 82	anbi12d 618	. . . . . . . . . . . . . 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)))))
84		rneq 5546	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
85	84	unieqd 4633	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → ∪ ran 𝑓 = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
86	85	fveq2d 6406	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (vol‘∪ ran 𝑓) = (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))))
87	84	imaeq2d 5670	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))))
88	87	supeq1d 8585	. . . . . . . . . . . . . . 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*, < ))
89	86, 88	eqeq12d 2817	. . . . . . . . . . . . . 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*, < )))
90	83, 89	imbi12d 335	. . . . . . . . . . . . 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*, < ))))
91		volsupnfl.0	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))
92	77, 90, 91	vtocl 3448	. . . . . . . . . . . 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*, < ))
93	55, 75, 92	sylancl 576	. . . . . . . . . . 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*, < ))
94		df-iun 4707	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥)}
95		eluzfz2 12566	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (ℤ≥‘1) → 𝑥 ∈ (1...𝑥))
96	95, 37	eleq2s 2899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
97		fveq2 6402	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑥 → (𝑔‘𝑙) = (𝑔‘𝑥))
98	97	eleq2d 2867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑥 → (𝑛 ∈ (𝑔‘𝑙) ↔ 𝑛 ∈ (𝑔‘𝑥)))
99	98	rspcev 3498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙))
100	96, 99	sylan 571	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙))
101		oveq2 6876	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
102	101	rexeqdv 3330	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙)))
103	102	rspcev 3498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔‘𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
104	100, 103	syldan 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔‘𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
105	104	rexlimiva 3212	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
106		ssrexv 3858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1...𝑚) ⊆ ℕ → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙)))
107	38, 106	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙))
108	98	cbvrexv 3357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔‘𝑙) ↔ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
109	107, 108	sylib 209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
110	109	rexlimivw 3213	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥))
111	105, 110	impbii 200	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
112		eliun 4709	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
113	112	rexbii 3225	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔‘𝑙))
114	111, 113	bitr4i 269	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
115	114	abbii 2919	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔‘𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
116	94, 115	eqtri 2824	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
117		df-iun 4707	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑚 ∈ ℕ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 ∈ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)}
118		ovex 6900	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑚) ∈ V
119	118, 63	iunex 7371	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ V
120	119	dfiun3 5575	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑚 ∈ ℕ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
121	116, 117, 120	3eqtr2i 2830	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))
122		fofn 6327	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
123		fniunfv 6723	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 Fn ℕ → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran 𝑔)
124	122, 123	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ ran 𝑔)
125		forn 6328	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ–onto→𝐴 → ran 𝑔 = 𝐴)
126	125	unieqd 4633	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ ran 𝑔 = ∪ 𝐴)
127	124, 126	eqtrd 2836	. . . . . . . . . . . . . 14 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ 𝑥 ∈ ℕ (𝑔‘𝑥) = ∪ 𝐴)
128	121, 127	syl5eqr 2850	. . . . . . . . . . . . 13 ⊢ (𝑔:ℕ–onto→𝐴 → ∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = ∪ 𝐴)
129	128	fveq2d 6406	. . . . . . . . . . . 12 ⊢ (𝑔:ℕ–onto→𝐴 → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol‘∪ 𝐴))
130	129	adantr 468	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol‘∪ 𝐴))
131		rnco2 5850	. . . . . . . . . . . . . 14 ⊢ ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
132		eqidd 2803	. . . . . . . . . . . . . . . . 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)))
133		volf 23504	. . . . . . . . . . . . . . . . . . 19 ⊢ vol:dom vol⟶(0[,]+∞)
134	133	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
135	134	feqmptd 6464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
136		fveq2 6402	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) → (vol‘𝑛) = (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
137	54, 132, 135, 136	fmptco 6613	. . . . . . . . . . . . . . . 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))))
138		mblvol 23505	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol → (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
139	54, 138	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)))
140		mblss 23506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
141	140	adantr 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
142	25	eqeq1d 2804	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
143		0re 10321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℝ
144		eleq1a 2876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
145	143, 144	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
146	142, 145	syl6bi 244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
147	146	imp 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
148	141, 147	jca 503	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
149	148	ralimi 3136	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150	149	adantl 469	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151		ssid 3814	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ℕ ⊆ ℕ
152		sseq1 3817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑔‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔‘𝑙) ⊆ ℝ))
153		fveq2 6402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑔‘𝑙) → (vol*‘𝑥) = (vol*‘(𝑔‘𝑙)))
154	153	eleq1d 2866	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑔‘𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
155	152, 154	anbi12d 618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝑔‘𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
156	155	ralima 6717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
157	122, 151, 156	sylancl 576	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
158		foima 6330	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑔:ℕ–onto→𝐴 → (𝑔 “ ℕ) = 𝐴)
159	158	raleqdv 3329	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
160	157, 159	bitr3d 272	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161	160	adantr 468	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
162	150, 161	mpbird 248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ))
163		ssralv 3857	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ ((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ) → ∀𝑙 ∈ (1...𝑚)((𝑔‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑔‘𝑙)) ∈ ℝ)))
164	38, 162, 163	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))
165	164	adantr 468	. . . . . . . . . . . . . . . . . . . . . 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))
166		ovolfiniun 23476	. . . . . . . . . . . . . . . . . . . . . 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)))
167	35, 165, 166	sylancr 577	. . . . . . . . . . . . . . . . . . . . 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)))
168		mblvol 23505	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔‘𝑙) ∈ dom vol → (vol‘(𝑔‘𝑙)) = (vol*‘(𝑔‘𝑙)))
169	48, 168	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔‘𝑙)) = (vol*‘(𝑔‘𝑙)))
170	44	simprd 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔‘𝑙) ∈ 𝐴) → (vol‘(𝑔‘𝑙)) = 0)
171	40, 170	sylan2 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ)) → (vol‘(𝑔‘𝑙)) = 0)
172	171	ancoms 448	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ 𝑙 ∈ ℕ) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔‘𝑙)) = 0)
173	172	an32s 634	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔‘𝑙)) = 0)
174	169, 173	eqtr3d 2838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔‘𝑙)) = 0)
175	174	ralrimiva 3150	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔‘𝑙)) = 0)
176		ssralv 3857	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (vol*‘(𝑔‘𝑙)) = 0 → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0))
177	38, 175, 176	mpsyl 68	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
178	177	adantr 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
179	178	sumeq2d 14649	. . . . . . . . . . . . . . . . . . . . . 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)
180	35	olci 884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑚) ⊆ (ℤ≥‘1) ∨ (1...𝑚) ∈ Fin)
181		sumz 14670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑚) ⊆ (ℤ≥‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑙 ∈ (1...𝑚)0 = 0)
182	180, 181	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑙 ∈ (1...𝑚)0 = 0
183	179, 182	syl6eq 2852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔‘𝑙)) = 0)
184	167, 183	breqtrd 4863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ≤ 0)
185		mblss 23506	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘𝑙) ∈ dom vol → (𝑔‘𝑙) ⊆ ℝ)
186	185	ralimi 3136	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
187	51, 186	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
188		iunss 4746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ ↔ ∀𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
189	187, 188	sylibr 225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
190	189	adantr 468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ)
191		ovolge0 23456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ → 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
192	190, 191	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)))
193		ovolcl 23453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙) ⊆ ℝ → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
194	189, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
195	194	adantr 468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ∈ ℝ*)
196		0xr 10365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ*
197		xrletri3 12197	. . . . . . . . . . . . . . . . . . . . 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)))))
198	195, 196, 197	sylancl 576	. . . . . . . . . . . . . . . . . . . 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)))))
199	184, 192, 198	mpbir2and 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0)
200	139, 199	eqtrd 2836	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) = 0)
201	200	mpteq2dva 4931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (𝑚 ∈ ℕ ↦ 0))
202		fconstmpt 5357	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
203	201, 202	syl6eqr 2854	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}))
204	137, 203	eqtrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = (ℕ × {0}))
205		frn 6256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ⊆ dom vol)
206		ffn 6250	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
207	133, 206	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ vol Fn dom vol
208	119, 61	fnmpti 6227	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) Fn ℕ
209		fnco 6204	. . . . . . . . . . . . . . . . . 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)
210	207, 208, 209	mp3an12 1568	. . . . . . . . . . . . . . . . 17 ⊢ (ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙)) ⊆ dom vol → (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) Fn ℕ)
211	55, 205, 210	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)
212		1nn 11310	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ
213	212	ne0ii 4119	. . . . . . . . . . . . . . . 16 ⊢ ℕ ≠ ∅
214		fconst5 6690	. . . . . . . . . . . . . . . 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}))
215	211, 213, 214	sylancl 576	. . . . . . . . . . . . . . 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}))
216	204, 215	mpbid 223	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ran (vol ∘ (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0})
217	131, 216	syl5eqr 2850	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ ∪ 𝑙 ∈ (1...𝑚)(𝑔‘𝑙))) = {0})
218	217	supeq1d 8585	. . . . . . . . . . . 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*, < ))
219		xrltso 12184	. . . . . . . . . . . . 13 ⊢ < Or ℝ*
220		supsn 8611	. . . . . . . . . . . . 13 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
221	219, 196, 220	mp2an 675	. . . . . . . . . . . 12 ⊢ sup({0}, ℝ*, < ) = 0
222	218, 221	syl6eq 2852	. . . . . . . . . . 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)
223	93, 130, 222	3eqtr3rd 2845	. . . . . . . . . 10 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘∪ 𝐴))
224	223	ex 399	. . . . . . . . 9 ⊢ (𝑔:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 0 = (vol‘∪ 𝐴)))
225	34, 224	syl5 34	. . . . . . . 8 ⊢ (𝑔:ℕ–onto→𝐴 → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
226	225	exlimiv 2020	. . . . . . 7 ⊢ (∃𝑔 𝑔:ℕ–onto→𝐴 → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
227	18, 226	syl 17	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → 0 = (vol‘∪ 𝐴)))
228	227	expimpd 443	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴)))
229	11, 228	pm2.61ine 3057	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol‘∪ 𝐴))
230		renepnf 10366	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
231	143, 230	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
232		fveq2 6402	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol‘∪ 𝐴) = (vol‘ℝ))
233		rembl 23515	. . . . . . . . 9 ⊢ ℝ ∈ dom vol
234		mblvol 23505	. . . . . . . . 9 ⊢ (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
235	233, 234	ax-mp 5	. . . . . . . 8 ⊢ (vol‘ℝ) = (vol*‘ℝ)
236		ovolre 23500	. . . . . . . 8 ⊢ (vol*‘ℝ) = +∞
237	235, 236	eqtri 2824	. . . . . . 7 ⊢ (vol‘ℝ) = +∞
238	232, 237	syl6eq 2852	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol‘∪ 𝐴) = +∞)
239	231, 238	neeqtrrd 3048	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol‘∪ 𝐴))
240	239	necon2i 3008	. . . 4 ⊢ (0 = (vol‘∪ 𝐴) → ∪ 𝐴 ≠ ℝ)
241	229, 240	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → ∪ 𝐴 ≠ ℝ)
242	241	expr 446	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
243		eqimss 3848	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
244	243	necon3bi 3000	. 2 ⊢ (¬ ∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ)
245	242, 244	pm2.61d1 172	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 865   = wceq 1637  ∃wex 1859   ∈ wcel 2155  {cab 2788   ≠ wne 2974  ∀wral 3092  ∃wrex 3093  Vcvv 3387   ⊆ wss 3763  ∅c0 4110  {csn 4364  ∪ cuni 4623  ∪ ciun 4705   class class class wbr 4837   ↦ cmpt 4916   Or wor 5225   × cxp 5303  dom cdm 5305  ran crn 5306   “ cima 5308   ∘ ccom 5309   Fn wfn 6090  ⟶wf 6091  –onto→wfo 6093  ‘cfv 6095  (class class class)co 6868   ≼ cdom 8184   ≺ csdm 8185  Fincfn 8186  supcsup 8579  ℝcr 10214  0cc0 10215  1c1 10216   + caddc 10218  +∞cpnf 10350  ℝ^*cxr 10352   < clt 10353   ≤ cle 10354  ℕcn 11299  ℤ_≥cuz 11898  [,]cicc 12390  ...cfz 12543  Σcsu 14633  vol*covol 23437  volcvol 23438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-inf2 8779  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292  ax-pre-sup 10293

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-isom 6104  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-of 7121  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-2o 7791  df-oadd 7794  df-er 7973  df-map 8088  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-fi 8550  df-sup 8581  df-inf 8582  df-oi 8648  df-card 9042  df-cda 9269  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-div 10964  df-nn 11300  df-2 11358  df-3 11359  df-n0 11554  df-z 11638  df-uz 11899  df-q 12002  df-rp 12041  df-xneg 12156  df-xadd 12157  df-xmul 12158  df-ioo 12391  df-ico 12393  df-icc 12394  df-fz 12544  df-fzo 12684  df-fl 12811  df-seq 13019  df-exp 13078  df-hash 13332  df-cj 14056  df-re 14057  df-im 14058  df-sqrt 14192  df-abs 14193  df-clim 14436  df-sum 14634  df-rest 16282  df-topgen 16303  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20906  df-topon 20923  df-bases 20958  df-cmp 21398  df-ovol 23439  df-vol 23440

This theorem is referenced by: (None)