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Theorem volsupnfl 34411
Description: volsup 23876 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 4717 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4736 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2825 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6501 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 23859 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 23850 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 23813 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2797 . . . . . . 7 (vol‘∅) = 0
104, 9syl6req 2826 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8311 . . . . . . . . . . 11 Rel ≼
1312brrelex1i 5455 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 8441 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 472 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 8463 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 580 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4740 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 615 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3115 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 270 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 23812 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24 nulmbl 23855 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25 mblvol 23850 . . . . . . . . . . . . . . . 16 (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26 eqtr 2794 . . . . . . . . . . . . . . . . 17 (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
2726expcom 406 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
3024, 29jcai 509 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3123, 30syldan 583 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3231ralimi 3105 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3322, 32sylbi 209 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3433ancoms 451 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35 fzfi 13154 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ Fin
36 fzssuz 12763 . . . . . . . . . . . . . . . . 17 (1...𝑚) ⊆ (ℤ‘1)
37 nnuz 12094 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
3836, 37sseqtr4i 3889 . . . . . . . . . . . . . . . 16 (1...𝑚) ⊆ ℕ
39 fof 6417 . . . . . . . . . . . . . . . . . . . 20 (𝑔:ℕ–onto𝐴𝑔:ℕ⟶𝐴)
4039ffvelrnda 6675 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) → (𝑔𝑙) ∈ 𝐴)
41 eleq1 2848 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → (𝑥 ∈ dom vol ↔ (𝑔𝑙) ∈ dom vol))
42 fveqeq2 6506 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔𝑙)) = 0))
4341, 42anbi12d 622 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑔𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0)))
4443rspccva 3529 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0))
4544simpld 487 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (𝑔𝑙) ∈ dom vol)
4645ancoms 451 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑙) ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4740, 46sylan 572 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4847an32s 640 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔𝑙) ∈ dom vol)
4948ralrimiva 3127 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol)
50 ssralv 3918 . . . . . . . . . . . . . . . 16 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol))
5138, 49, 50mpsyl 68 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
52 finiunmbl 23864 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5335, 51, 52sylancr 579 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5453adantr 473 . . . . . . . . . . . . 13 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5554fmpttd 6701 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol)
56 fzssp1 12765 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ (1...(𝑛 + 1))
57 iunss1 4802 . . . . . . . . . . . . . . 15 ((1...𝑛) ⊆ (1...(𝑛 + 1)) → 𝑙 ∈ (1...𝑛)(𝑔𝑙) ⊆ 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 𝑙 ∈ (1...𝑛)(𝑔𝑙) ⊆ 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙)
59 oveq2 6983 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
6059iuneq1d 4815 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
61 eqid 2773 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
62 ovex 7007 . . . . . . . . . . . . . . . . 17 (1...𝑛) ∈ V
63 fvex 6510 . . . . . . . . . . . . . . . . 17 (𝑔𝑙) ∈ V
6462, 63iunex 7480 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑛)(𝑔𝑙) ∈ V
6560, 61, 64fvmpt 6594 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
66 peano2nn 11452 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
67 oveq2 6983 . . . . . . . . . . . . . . . . . 18 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
6867iuneq1d 4815 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 + 1) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
69 ovex 7007 . . . . . . . . . . . . . . . . . 18 (1...(𝑛 + 1)) ∈ V
7069, 63iunex 7480 . . . . . . . . . . . . . . . . 17 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙) ∈ V
7168, 61, 70fvmpt 6594 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
7266, 71syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
7365, 72sseq12d 3885 . . . . . . . . . . . . . 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)))
7458, 73mpbiri 250 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
7574rgen 3093 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))
76 nnex 11445 . . . . . . . . . . . . . 14 ℕ ∈ V
7776mptex 6811 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ V
78 feq1 6323 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol))
79 fveq1 6496 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛))
80 fveq1 6496 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
8179, 80sseq12d 3885 . . . . . . . . . . . . . . . 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))))
8281ralbidv 3142 . . . . . . . . . . . . . . 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))))
8378, 82anbi12d 622 . . . . . . . . . . . . . 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)))))
84 rneq 5647 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8584unieqd 4719 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8685fveq2d 6501 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol‘ ran 𝑓) = (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8784imaeq2d 5768 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8887supeq1d 8704 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → sup((vol “ ran 𝑓), ℝ*, < ) = sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ))
8986, 88eqeq12d 2788 . . . . . . . . . . . . . 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*, < )))
9083, 89imbi12d 337 . . . . . . . . . . . . 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*, < ))))
91 volsupnfl.0 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))
9277, 90, 91vtocl 3473 . . . . . . . . . . . 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*, < ))
9355, 75, 92sylancl 578 . . . . . . . . . . 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*, < ))
94 df-iun 4791 . . . . . . . . . . . . . . . 16 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)}
95 eluzfz2 12730 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ (1...𝑥))
9695, 37eleq2s 2879 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
97 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑥 → (𝑔𝑙) = (𝑔𝑥))
9897eleq2d 2846 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑥 → (𝑛 ∈ (𝑔𝑙) ↔ 𝑛 ∈ (𝑔𝑥)))
9998rspcev 3530 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
10096, 99sylan 572 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
101 oveq2 6983 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
102101rexeqdv 3351 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)))
103102rspcev 3530 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
104100, 103syldan 583 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
105104rexlimiva 3221 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
106 ssrexv 3919 . . . . . . . . . . . . . . . . . . . . . 22 ((1...𝑚) ⊆ ℕ → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙)))
10738, 106ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙))
10898cbvrexv 3379 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙) ↔ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
109107, 108sylib 210 . . . . . . . . . . . . . . . . . . . 20 (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
110109rexlimivw 3222 . . . . . . . . . . . . . . . . . . 19 (∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
111105, 110impbii 201 . . . . . . . . . . . . . . . . . 18 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
112 eliun 4793 . . . . . . . . . . . . . . . . . . 19 (𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
113112rexbii 3189 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
114111, 113bitr4i 270 . . . . . . . . . . . . . . . . 17 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙))
115114abbii 2839 . . . . . . . . . . . . . . . 16 {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
11694, 115eqtri 2797 . . . . . . . . . . . . . . 15 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
117 df-iun 4791 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
118 ovex 7007 . . . . . . . . . . . . . . . . 17 (1...𝑚) ∈ V
119118, 63iunex 7480 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ V
120119dfiun3 5677 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
121116, 117, 1203eqtr2i 2803 . . . . . . . . . . . . . 14 𝑥 ∈ ℕ (𝑔𝑥) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
122 fofn 6419 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
123 fniunfv 6830 . . . . . . . . . . . . . . . 16 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
124122, 123syl 17 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
125 forn 6420 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 → ran 𝑔 = 𝐴)
126125unieqd 4719 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 ran 𝑔 = 𝐴)
127124, 126eqtrd 2809 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = 𝐴)
128121, 127syl5eqr 2823 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 𝐴)
129128fveq2d 6501 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
130129adantr 473 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
131 rnco2 5943 . . . . . . . . . . . . . 14 ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
132 eqidd 2774 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
133 volf 23849 . . . . . . . . . . . . . . . . . . 19 vol:dom vol⟶(0[,]+∞)
134133a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
135134feqmptd 6561 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
136 fveq2 6497 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑙 ∈ (1...𝑚)(𝑔𝑙) → (vol‘𝑛) = (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
13754, 132, 135, 136fmptco 6713 . . . . . . . . . . . . . . . 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))))
138 mblvol 23850 . . . . . . . . . . . . . . . . . . . 20 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
13954, 138syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
140 mblss 23851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
141140adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
14225eqeq1d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
143 0re 10440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℝ
144 eleq1a 2856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
145143, 144ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
146142, 145syl6bi 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
147146imp 398 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
148141, 147jca 504 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
149148ralimi 3105 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150149adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151 ssid 3874 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℕ ⊆ ℕ
152 sseq1 3877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔𝑙) ⊆ ℝ))
153 fveq2 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑔𝑙) → (vol*‘𝑥) = (vol*‘(𝑔𝑙)))
154153eleq1d 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔𝑙)) ∈ ℝ))
155152, 154anbi12d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝑔𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
156155ralima 6824 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
157122, 151, 156sylancl 578 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
158 foima 6422 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
159158raleqdv 3350 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
160157, 159bitr3d 273 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161160adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
162150, 161mpbird 249 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
163 ssralv 3918 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
16438, 162, 163mpsyl 68 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
165164adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
166 ovolfiniun 23821 . . . . . . . . . . . . . . . . . . . . . 22 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)))
16735, 165, 166sylancr 579 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)))
168 mblvol 23850 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔𝑙) ∈ dom vol → (vol‘(𝑔𝑙)) = (vol*‘(𝑔𝑙)))
16948, 168syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = (vol*‘(𝑔𝑙)))
17044simprd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (vol‘(𝑔𝑙)) = 0)
17140, 170sylan2 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto𝐴𝑙 ∈ ℕ)) → (vol‘(𝑔𝑙)) = 0)
172171ancoms 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔𝑙)) = 0)
173172an32s 640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = 0)
174169, 173eqtr3d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔𝑙)) = 0)
175174ralrimiva 3127 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0)
176 ssralv 3918 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1...𝑚) ⊆ ℕ → (∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0 → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0))
17738, 175, 176mpsyl 68 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
178177adantr 473 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
179178sumeq2d 14918 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = Σ𝑙 ∈ (1...𝑚)0)
18035olci 853 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin)
181 sumz 14938 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑙 ∈ (1...𝑚)0 = 0)
182180, 181ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 Σ𝑙 ∈ (1...𝑚)0 = 0
183179, 182syl6eq 2825 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
184167, 183breqtrd 4952 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0)
185 mblss 23851 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑙) ∈ dom vol → (𝑔𝑙) ⊆ ℝ)
186185ralimi 3105 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
18751, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
188 iunss 4832 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ ↔ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
189187, 188sylibr 226 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
190189adantr 473 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
191 ovolge0 23801 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ → 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
192190, 191syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
193 ovolcl 23798 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
194189, 193syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
195194adantr 473 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
196 0xr 10486 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ*
197 xrletri3 12363 . . . . . . . . . . . . . . . . . . . . 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)))))
198195, 196, 197sylancl 578 . . . . . . . . . . . . . . . . . . . 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)))))
199184, 192, 198mpbir2and 701 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
200139, 199eqtrd 2809 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
201200mpteq2dva 5019 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ 0))
202 fconstmpt 5461 . . . . . . . . . . . . . . . . 17 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
203201, 202syl6eqr 2827 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
204137, 203eqtrd 2809 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
205 frn 6348 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol)
206 ffn 6342 . . . . . . . . . . . . . . . . . . 19 (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
207133, 206ax-mp 5 . . . . . . . . . . . . . . . . . 18 vol Fn dom vol
208119, 61fnmpti 6319 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) Fn ℕ
209 fnco 6296 . . . . . . . . . . . . . . . . . 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)
210207, 208, 209mp3an12 1431 . . . . . . . . . . . . . . . . 17 (ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ)
21155, 205, 2103syl 18 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) Fn ℕ)
212 1nn 11451 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ
213212ne0ii 4184 . . . . . . . . . . . . . . . 16 ℕ ≠ ∅
214 fconst5 6794 . . . . . . . . . . . . . . . 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}))
215211, 213, 214sylancl 578 . . . . . . . . . . . . . . 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}))
216204, 215mpbid 224 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
217131, 216syl5eqr 2823 . . . . . . . . . . . . 13 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
218217supeq1d 8704 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ) = sup({0}, ℝ*, < ))
219 xrltso 12350 . . . . . . . . . . . . 13 < Or ℝ*
220 supsn 8730 . . . . . . . . . . . . 13 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
221219, 196, 220mp2an 680 . . . . . . . . . . . 12 sup({0}, ℝ*, < ) = 0
222218, 221syl6eq 2825 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → sup((vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))), ℝ*, < ) = 0)
22393, 130, 2223eqtr3rd 2818 . . . . . . . . . 10 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘ 𝐴))
224223ex 405 . . . . . . . . 9 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 0 = (vol‘ 𝐴)))
22534, 224syl5 34 . . . . . . . 8 (𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
226225exlimiv 1890 . . . . . . 7 (∃𝑔 𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
22718, 226syl 17 . . . . . 6 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
228227expimpd 446 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
22911, 228pm2.61ine 3046 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
230 renepnf 10487 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
231143, 230mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
232 fveq2 6497 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
233 rembl 23860 . . . . . . . . 9 ℝ ∈ dom vol
234 mblvol 23850 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
235233, 234ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
236 ovolre 23845 . . . . . . . 8 (vol*‘ℝ) = +∞
237235, 236eqtri 2797 . . . . . . 7 (vol‘ℝ) = +∞
238232, 237syl6eq 2825 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
239231, 238neeqtrrd 3036 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
240239necon2i 2996 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
241229, 240syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
242241expr 449 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
243 eqimss 3908 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
244243necon3bi 2988 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
245242, 244pm2.61d1 173 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wo 834   = wceq 1508  wex 1743  wcel 2051  {cab 2753  wne 2962  wral 3083  wrex 3084  Vcvv 3410  wss 3824  c0 4173  {csn 4436   cuni 4709   ciun 4789   class class class wbr 4926  cmpt 5005   Or wor 5322   × cxp 5402  dom cdm 5404  ran crn 5405  cima 5407  ccom 5408   Fn wfn 6181  wf 6182  ontowfo 6184  cfv 6186  (class class class)co 6975  cdom 8303  csdm 8304  Fincfn 8305  supcsup 8698  cr 10333  0cc0 10334  1c1 10335   + caddc 10337  +∞cpnf 10470  *cxr 10472   < clt 10473  cle 10474  cn 11438  cuz 12057  [,]cicc 12556  ...cfz 12707  Σcsu 14902  vol*covol 23782  volcvol 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-fi 8669  df-sup 8700  df-inf 8701  df-oi 8768  df-dju 9123  df-card 9161  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-q 12162  df-rp 12204  df-xneg 12323  df-xadd 12324  df-xmul 12325  df-ioo 12557  df-ico 12559  df-icc 12560  df-fz 12708  df-fzo 12849  df-fl 12976  df-seq 13184  df-exp 13244  df-hash 13505  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-clim 14705  df-sum 14903  df-rest 16551  df-topgen 16572  df-psmet 20255  df-xmet 20256  df-met 20257  df-bl 20258  df-mopn 20259  df-top 21222  df-topon 21239  df-bases 21274  df-cmp 21715  df-ovol 23784  df-vol 23785
This theorem is referenced by: (None)
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