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Theorem volsupnfl 33584
Description: volsup 23370 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 (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))
Assertion
Ref Expression
volsupnfl ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 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.
StepHypRef Expression
1 unieq 4476 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4497 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2701 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6233 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 23353 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 23344 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 23307 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2673 . . . . . . 7 (vol‘∅) = 0
104, 9syl6req 2702 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8003 . . . . . . . . . . 11 Rel ≼
1312brrelexi 5192 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 8130 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 503 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 8152 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 702 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4501 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 731 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3093 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 267 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 23306 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24 nulmbl 23349 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25 mblvol 23344 . . . . . . . . . . . . . . . 16 (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26 eqtr 2670 . . . . . . . . . . . . . . . . 17 (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
2726expcom 450 . . . . . . . . . . . . . . . 16 ((vol*‘𝑥) = 0 → ((vol‘𝑥) = (vol*‘𝑥) → (vol‘𝑥) = 0))
2825, 27syl5 34 . . . . . . . . . . . . . . 15 ((vol*‘𝑥) = 0 → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
2928adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
3024, 29jcai 558 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3123, 30syldan 486 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3231ralimi 2981 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3322, 32sylbi 207 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3433ancoms 468 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35 fzfi 12811 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ Fin
36 fzssuz 12420 . . . . . . . . . . . . . . . . 17 (1...𝑚) ⊆ (ℤ‘1)
37 nnuz 11761 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
3836, 37sseqtr4i 3671 . . . . . . . . . . . . . . . 16 (1...𝑚) ⊆ ℕ
39 fof 6153 . . . . . . . . . . . . . . . . . . . 20 (𝑔:ℕ–onto𝐴𝑔:ℕ⟶𝐴)
4039ffvelrnda 6399 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) → (𝑔𝑙) ∈ 𝐴)
41 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → (𝑥 ∈ dom vol ↔ (𝑔𝑙) ∈ dom vol))
42 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑔𝑙) → (vol‘𝑥) = (vol‘(𝑔𝑙)))
4342eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔𝑙)) = 0))
4441, 43anbi12d 747 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑔𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0)))
4544rspccva 3339 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0))
4645simpld 474 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (𝑔𝑙) ∈ dom vol)
4746ancoms 468 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑙) ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4840, 47sylan 487 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4948an32s 863 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔𝑙) ∈ dom vol)
5049ralrimiva 2995 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol)
51 ssralv 3699 . . . . . . . . . . . . . . . 16 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol))
5238, 50, 51mpsyl 68 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
53 finiunmbl 23358 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5435, 52, 53sylancr 696 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5554adantr 480 . . . . . . . . . . . . 13 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
56 eqid 2651 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
5755, 56fmptd 6425 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol)
58 fzssp1 12422 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ (1...(𝑛 + 1))
59 iunss1 4564 . . . . . . . . . . . . . . 15 ((1...𝑛) ⊆ (1...(𝑛 + 1)) → 𝑙 ∈ (1...𝑛)(𝑔𝑙) ⊆ 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
6058, 59ax-mp 5 . . . . . . . . . . . . . 14 𝑙 ∈ (1...𝑛)(𝑔𝑙) ⊆ 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙)
61 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
6261iuneq1d 4577 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
63 ovex 6718 . . . . . . . . . . . . . . . . 17 (1...𝑛) ∈ V
64 fvex 6239 . . . . . . . . . . . . . . . . 17 (𝑔𝑙) ∈ V
6563, 64iunex 7189 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑛)(𝑔𝑙) ∈ V
6662, 56, 65fvmpt 6321 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
67 peano2nn 11070 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
68 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
6968iuneq1d 4577 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 + 1) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
70 ovex 6718 . . . . . . . . . . . . . . . . . 18 (1...(𝑛 + 1)) ∈ V
7170, 64iunex 7189 . . . . . . . . . . . . . . . . 17 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙) ∈ V
7269, 56, 71fvmpt 6321 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
7367, 72syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
7466, 73sseq12d 3667 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)) ↔ 𝑙 ∈ (1...𝑛)(𝑔𝑙) ⊆ 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙)))
7560, 74mpbiri 248 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
7675rgen 2951 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))
77 nnex 11064 . . . . . . . . . . . . . 14 ℕ ∈ V
7877mptex 6527 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ V
79 feq1 6064 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol))
80 fveq1 6228 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛))
81 fveq1 6228 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
8280, 81sseq12d 3667 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ((𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1)) ↔ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))))
8382ralbidv 3015 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (∀𝑛 ∈ ℕ (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1)) ↔ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))))
8479, 83anbi12d 747 . . . . . . . . . . . . . 14 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1))) ↔ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))))
85 rneq 5383 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8685unieqd 4478 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8786fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol‘ ran 𝑓) = (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8885imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8988supeq1d 8393 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → sup((vol “ ran 𝑓), ℝ*, < ) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ))
9087, 89eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ((vol‘ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ) ↔ (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < )))
9184, 90imbi12d 333 . . . . . . . . . . . . 13 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < )) ↔ (((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ))))
92 volsupnfl.0 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))
9378, 91, 92vtocl 3290 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ))
9457, 76, 93sylancl 695 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ))
95 df-iun 4554 . . . . . . . . . . . . . . . 16 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)}
96 eluzfz2 12387 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ (1...𝑥))
9796, 37eleq2s 2748 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
98 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑥 → (𝑔𝑙) = (𝑔𝑥))
9998eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑥 → (𝑛 ∈ (𝑔𝑙) ↔ 𝑛 ∈ (𝑔𝑥)))
10099rspcev 3340 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
10197, 100sylan 487 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
102 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
103102rexeqdv 3175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)))
104103rspcev 3340 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
105101, 104syldan 486 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
106105rexlimiva 3057 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
107 ssrexv 3700 . . . . . . . . . . . . . . . . . . . . . 22 ((1...𝑚) ⊆ ℕ → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙)))
10838, 107ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙))
10999cbvrexv 3202 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙) ↔ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
110108, 109sylib 208 . . . . . . . . . . . . . . . . . . . 20 (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
111110rexlimivw 3058 . . . . . . . . . . . . . . . . . . 19 (∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
112106, 111impbii 199 . . . . . . . . . . . . . . . . . 18 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
113 eliun 4556 . . . . . . . . . . . . . . . . . . 19 (𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
114113rexbii 3070 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
115112, 114bitr4i 267 . . . . . . . . . . . . . . . . 17 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙))
116115abbii 2768 . . . . . . . . . . . . . . . 16 {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
11795, 116eqtri 2673 . . . . . . . . . . . . . . 15 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
118 df-iun 4554 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
119 ovex 6718 . . . . . . . . . . . . . . . . 17 (1...𝑚) ∈ V
120119, 64iunex 7189 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ V
121120dfiun3 5412 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
122117, 118, 1213eqtr2i 2679 . . . . . . . . . . . . . 14 𝑥 ∈ ℕ (𝑔𝑥) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
123 fofn 6155 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
124 fniunfv 6545 . . . . . . . . . . . . . . . 16 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
125123, 124syl 17 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
126 forn 6156 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 → ran 𝑔 = 𝐴)
127126unieqd 4478 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 ran 𝑔 = 𝐴)
128125, 127eqtrd 2685 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = 𝐴)
129122, 128syl5eqr 2699 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 𝐴)
130129fveq2d 6233 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
131130adantr 480 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
132 rnco2 5680 . . . . . . . . . . . . . 14 ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
133 eqidd 2652 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
134 volf 23343 . . . . . . . . . . . . . . . . . . 19 vol:dom vol⟶(0[,]+∞)
135134a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
136135feqmptd 6288 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
137 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑙 ∈ (1...𝑚)(𝑔𝑙) → (vol‘𝑛) = (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
13855, 133, 136, 137fmptco 6436 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
139 mblvol 23344 . . . . . . . . . . . . . . . . . . . 20 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
14055, 139syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
141 mblss 23345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
142141adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
14325eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
144 0re 10078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℝ
145 eleq1a 2725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
146144, 145ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
147143, 146syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
148147imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
149142, 148jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150149ralimi 2981 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151150adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
152 ssid 3657 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℕ ⊆ ℕ
153 sseq1 3659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔𝑙) ⊆ ℝ))
154 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑔𝑙) → (vol*‘𝑥) = (vol*‘(𝑔𝑙)))
155154eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔𝑙)) ∈ ℝ))
156153, 155anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝑔𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
157156ralima 6538 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
158123, 152, 157sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
159 foima 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
160159raleqdv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161158, 160bitr3d 270 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
162161adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
163151, 162mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
164 ssralv 3699 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
16538, 163, 164mpsyl 68 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
166165adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
167 ovolfiniun 23315 . . . . . . . . . . . . . . . . . . . . . 22 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)))
16835, 166, 167sylancr 696 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)))
169 mblvol 23344 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔𝑙) ∈ dom vol → (vol‘(𝑔𝑙)) = (vol*‘(𝑔𝑙)))
17049, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = (vol*‘(𝑔𝑙)))
17145simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (vol‘(𝑔𝑙)) = 0)
17240, 171sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto𝐴𝑙 ∈ ℕ)) → (vol‘(𝑔𝑙)) = 0)
173172ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔𝑙)) = 0)
174173an32s 863 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = 0)
175170, 174eqtr3d 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔𝑙)) = 0)
176175ralrimiva 2995 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0)
177 ssralv 3699 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0 → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0))
17838, 176, 177mpsyl 68 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
179178adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
180179sumeq2d 14476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = Σ𝑙 ∈ (1...𝑚)0)
18135olci 405 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin)
182 sumz 14497 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑙 ∈ (1...𝑚)0 = 0)
183181, 182ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 Σ𝑙 ∈ (1...𝑚)0 = 0
184180, 183syl6eq 2701 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
185168, 184breqtrd 4711 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0)
186 mblss 23345 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑙) ∈ dom vol → (𝑔𝑙) ⊆ ℝ)
187186ralimi 2981 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
18852, 187syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
189 iunss 4593 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ ↔ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
190188, 189sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
191190adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
192 ovolge0 23295 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ → 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
193191, 192syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
194 ovolcl 23292 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
195190, 194syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
196195adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
197 0xr 10124 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ*
198 xrletri3 12023 . . . . . . . . . . . . . . . . . . . . 21 (((vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0 ↔ ((vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0 ∧ 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))))
199196, 197, 198sylancl 695 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ((vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0 ↔ ((vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0 ∧ 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))))
200185, 193, 199mpbir2and 977 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
201140, 200eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
202201mpteq2dva 4777 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ 0))
203 fconstmpt 5197 . . . . . . . . . . . . . . . . 17 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
204202, 203syl6eqr 2703 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
205138, 204eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
206 frn 6091 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol)
207 ffn 6083 . . . . . . . . . . . . . . . . . . 19 (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
208134, 207ax-mp 5 . . . . . . . . . . . . . . . . . 18 vol Fn dom vol
209120, 56fnmpti 6060 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) Fn ℕ
210 fnco 6037 . . . . . . . . . . . . . . . . . 18 ((vol Fn dom vol ∧ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) Fn ℕ ∧ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ)
211208, 209, 210mp3an12 1454 . . . . . . . . . . . . . . . . 17 (ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ)
21257, 206, 2113syl 18 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ)
213 1nn 11069 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ
214213ne0ii 3956 . . . . . . . . . . . . . . . 16 ℕ ≠ ∅
215 fconst5 6512 . . . . . . . . . . . . . . . 16 (((vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ ∧ ℕ ≠ ∅) → ((vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}) ↔ ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0}))
216212, 214, 215sylancl 695 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ((vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}) ↔ ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0}))
217205, 216mpbid 222 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
218132, 217syl5eqr 2699 . . . . . . . . . . . . 13 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
219218supeq1d 8393 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ) = sup({0}, ℝ*, < ))
220 xrltso 12012 . . . . . . . . . . . . 13 < Or ℝ*
221 supsn 8419 . . . . . . . . . . . . 13 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
222220, 197, 221mp2an 708 . . . . . . . . . . . 12 sup({0}, ℝ*, < ) = 0
223219, 222syl6eq 2701 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ) = 0)
22494, 131, 2233eqtr3rd 2694 . . . . . . . . . 10 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘ 𝐴))
225224ex 449 . . . . . . . . 9 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 0 = (vol‘ 𝐴)))
22634, 225syl5 34 . . . . . . . 8 (𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
227226exlimiv 1898 . . . . . . 7 (∃𝑔 𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
22818, 227syl 17 . . . . . 6 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
229228expimpd 628 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
23011, 229pm2.61ine 2906 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
231 renepnf 10125 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
232144, 231mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
233 fveq2 6229 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
234 rembl 23354 . . . . . . . . 9 ℝ ∈ dom vol
235 mblvol 23344 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
236234, 235ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
237 ovolre 23339 . . . . . . . 8 (vol*‘ℝ) = +∞
238236, 237eqtri 2673 . . . . . . 7 (vol‘ℝ) = +∞
239233, 238syl6eq 2701 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
240232, 239neeqtrrd 2897 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
241240necon2i 2857 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
242230, 241syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
243242expr 642 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
244 eqimss 3690 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
245244necon3bi 2849 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
246243, 245pm2.61d1 171 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   Or wor 5063   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  cdom 7995  csdm 7996  Fincfn 7997  supcsup 8387  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cn 11058  cuz 11725  [,]cicc 12216  ...cfz 12364  Σcsu 14460  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280
This theorem is referenced by: (None)
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