Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  volsupnfl Structured version   Visualization version   GIF version

Theorem volsupnfl 36169
Description: volsup 24936 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 4877 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4897 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2789 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6847 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 24919 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 24910 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 24873 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2761 . . . . . . 7 (vol‘∅) = 0
104, 9eqtr2di 2790 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8892 . . . . . . . . . . 11 Rel ≼
1312brrelex1i 5689 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 9051 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 481 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 9075 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 589 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4901 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 625 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3111 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 278 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 24872 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24 nulmbl 24915 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25 mblvol 24910 . . . . . . . . . . . . . . . 16 (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26 eqtr 2756 . . . . . . . . . . . . . . . . 17 (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
2726expcom 415 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
3024, 29jcai 518 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3123, 30syldan 592 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3231ralimi 3083 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3322, 32sylbi 216 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3433ancoms 460 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35 fzfi 13883 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ Fin
36 fzssuz 13488 . . . . . . . . . . . . . . . . 17 (1...𝑚) ⊆ (ℤ‘1)
37 nnuz 12811 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
3836, 37sseqtrri 3982 . . . . . . . . . . . . . . . 16 (1...𝑚) ⊆ ℕ
39 fof 6757 . . . . . . . . . . . . . . . . . . . 20 (𝑔:ℕ–onto𝐴𝑔:ℕ⟶𝐴)
4039ffvelcdmda 7036 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) → (𝑔𝑙) ∈ 𝐴)
41 eleq1 2822 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → (𝑥 ∈ dom vol ↔ (𝑔𝑙) ∈ dom vol))
42 fveqeq2 6852 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔𝑙)) = 0))
4341, 42anbi12d 632 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑔𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0)))
4443rspccva 3579 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0))
4544simpld 496 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (𝑔𝑙) ∈ dom vol)
4645ancoms 460 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑙) ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4740, 46sylan 581 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4847an32s 651 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔𝑙) ∈ dom vol)
4948ralrimiva 3140 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol)
50 ssralv 4011 . . . . . . . . . . . . . . . 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 24924 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5335, 51, 52sylancr 588 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5453adantr 482 . . . . . . . . . . . . 13 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5554fmpttd 7064 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol)
56 fzssp1 13490 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ (1...(𝑛 + 1))
57 iunss1 4969 . . . . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
6059iuneq1d 4982 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
61 eqid 2733 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
62 ovex 7391 . . . . . . . . . . . . . . . . 17 (1...𝑛) ∈ V
63 fvex 6856 . . . . . . . . . . . . . . . . 17 (𝑔𝑙) ∈ V
6462, 63iunex 7902 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑛)(𝑔𝑙) ∈ V
6560, 61, 64fvmpt 6949 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
66 peano2nn 12170 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
67 oveq2 7366 . . . . . . . . . . . . . . . . . 18 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
6867iuneq1d 4982 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 + 1) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
69 ovex 7391 . . . . . . . . . . . . . . . . . 18 (1...(𝑛 + 1)) ∈ V
7069, 63iunex 7902 . . . . . . . . . . . . . . . . 17 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙) ∈ V
7168, 61, 70fvmpt 6949 . . . . . . . . . . . . . . . 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 3978 . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
7574rgen 3063 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))
76 nnex 12164 . . . . . . . . . . . . . 14 ℕ ∈ V
7776mptex 7174 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ V
78 feq1 6650 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol))
79 fveq1 6842 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛))
80 fveq1 6842 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
8179, 80sseq12d 3978 . . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . 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 5892 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8584unieqd 4880 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8685fveq2d 6847 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol‘ ran 𝑓) = (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8784imaeq2d 6014 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8887supeq1d 9387 . . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . 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 345 . . . . . . . . . . . . 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 3517 . . . . . . . . . . . 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 587 . . . . . . . . . . 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 4957 . . . . . . . . . . . . . . . 16 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)}
95 eluzfz2 13455 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ (1...𝑥))
9695, 37eleq2s 2852 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
97 fveq2 6843 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑥 → (𝑔𝑙) = (𝑔𝑥))
9897eleq2d 2820 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑥 → (𝑛 ∈ (𝑔𝑙) ↔ 𝑛 ∈ (𝑔𝑥)))
9998rspcev 3580 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
10096, 99sylan 581 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
101 oveq2 7366 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
102101rexeqdv 3313 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)))
103102rspcev 3580 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
104100, 103syldan 592 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
105104rexlimiva 3141 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
106 ssrexv 4012 . . . . . . . . . . . . . . . . . . . . . 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))
10898cbvrexvw 3225 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑙 ∈ ℕ 𝑛 ∈ (𝑔𝑙) ↔ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
109107, 108sylib 217 . . . . . . . . . . . . . . . . . . . 20 (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
110109rexlimivw 3145 . . . . . . . . . . . . . . . . . . 19 (∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) → ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥))
111105, 110impbii 208 . . . . . . . . . . . . . . . . . 18 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
112 eliun 4959 . . . . . . . . . . . . . . . . . . 19 (𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
113112rexbii 3094 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
114111, 113bitr4i 278 . . . . . . . . . . . . . . . . 17 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙))
115114abbii 2803 . . . . . . . . . . . . . . . 16 {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
11694, 115eqtri 2761 . . . . . . . . . . . . . . 15 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
117 df-iun 4957 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
118 ovex 7391 . . . . . . . . . . . . . . . . 17 (1...𝑚) ∈ V
119118, 63iunex 7902 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ V
120119dfiun3 5922 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
121116, 117, 1203eqtr2i 2767 . . . . . . . . . . . . . 14 𝑥 ∈ ℕ (𝑔𝑥) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
122 fofn 6759 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
123 fniunfv 7195 . . . . . . . . . . . . . . . 16 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
124122, 123syl 17 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
125 forn 6760 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 → ran 𝑔 = 𝐴)
126125unieqd 4880 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 ran 𝑔 = 𝐴)
127124, 126eqtrd 2773 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = 𝐴)
128121, 127eqtr3id 2787 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 𝐴)
129128fveq2d 6847 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
130129adantr 482 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
131 rnco2 6206 . . . . . . . . . . . . . 14 ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
132 eqidd 2734 . . . . . . . . . . . . . . . . 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 24909 . . . . . . . . . . . . . . . . . . 19 vol:dom vol⟶(0[,]+∞)
134133a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
135134feqmptd 6911 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
136 fveq2 6843 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑙 ∈ (1...𝑚)(𝑔𝑙) → (vol‘𝑛) = (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
13754, 132, 135, 136fmptco 7076 . . . . . . . . . . . . . . . 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 24910 . . . . . . . . . . . . . . . . . . . 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 24911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
141140adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
14225eqeq1d 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
143 0re 11162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℝ
144 eleq1a 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
145143, 144ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
146142, 145syl6bi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
147146imp 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
148141, 147jca 513 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
149148ralimi 3083 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150149adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151 ssid 3967 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℕ ⊆ ℕ
152 sseq1 3970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔𝑙) ⊆ ℝ))
153 fveq2 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑔𝑙) → (vol*‘𝑥) = (vol*‘(𝑔𝑙)))
154153eleq1d 2819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔𝑙)) ∈ ℝ))
155152, 154anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝑔𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
156155ralima 7189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
157122, 151, 156sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
158 foima 6762 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
159158raleqdv 3312 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
160157, 159bitr3d 281 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161160adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
163 ssralv 4011 . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
166 ovolfiniun 24881 . . . . . . . . . . . . . . . . . . . . . 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 588 . . . . . . . . . . . . . . . . . . . . 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 24910 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (vol‘(𝑔𝑙)) = 0)
17140, 170sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto𝐴𝑙 ∈ ℕ)) → (vol‘(𝑔𝑙)) = 0)
172171ancoms 460 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔𝑙)) = 0)
173172an32s 651 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = 0)
174169, 173eqtr3d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔𝑙)) = 0)
175174ralrimiva 3140 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0)
176 ssralv 4011 . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
179178sumeq2d 15592 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = Σ𝑙 ∈ (1...𝑚)0)
18035olci 865 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin)
181 sumz 15612 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑙 ∈ (1...𝑚)0 = 0)
182180, 181ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 Σ𝑙 ∈ (1...𝑚)0 = 0
183179, 182eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
184167, 183breqtrd 5132 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0)
185 mblss 24911 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑙) ∈ dom vol → (𝑔𝑙) ⊆ ℝ)
186185ralimi 3083 . . . . . . . . . . . . . . . . . . . . . . . 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 5006 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ ↔ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
189187, 188sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
190189adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
191 ovolge0 24861 . . . . . . . . . . . . . . . . . . . . 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 24858 . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
196 0xr 11207 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ*
197 xrletri3 13079 . . . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
200139, 199eqtrd 2773 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
201200mpteq2dva 5206 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ 0))
202 fconstmpt 5695 . . . . . . . . . . . . . . . . 17 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
203201, 202eqtr4di 2791 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
204137, 203eqtrd 2773 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
205 frn 6676 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol)
206 ffn 6669 . . . . . . . . . . . . . . . . . . 19 (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
207133, 206ax-mp 5 . . . . . . . . . . . . . . . . . 18 vol Fn dom vol
208119, 61fnmpti 6645 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) Fn ℕ
209 fnco 6619 . . . . . . . . . . . . . . . . . 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 1452 . . . . . . . . . . . . . . . . 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 12169 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ
213212ne0ii 4298 . . . . . . . . . . . . . . . 16 ℕ ≠ ∅
214 fconst5 7156 . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
217131, 216eqtr3id 2787 . . . . . . . . . . . . 13 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
218217supeq1d 9387 . . . . . . . . . . . 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 13066 . . . . . . . . . . . . 13 < Or ℝ*
220 supsn 9413 . . . . . . . . . . . . 13 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
221219, 196, 220mp2an 691 . . . . . . . . . . . 12 sup({0}, ℝ*, < ) = 0
222218, 221eqtrdi 2789 . . . . . . . . . . 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 2782 . . . . . . . . . 10 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘ 𝐴))
224223ex 414 . . . . . . . . 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 1934 . . . . . . 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 455 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
22911, 228pm2.61ine 3025 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
230 renepnf 11208 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
231143, 230mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
232 fveq2 6843 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
233 rembl 24920 . . . . . . . . 9 ℝ ∈ dom vol
234 mblvol 24910 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
235233, 234ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
236 ovolre 24905 . . . . . . . 8 (vol*‘ℝ) = +∞
237235, 236eqtri 2761 . . . . . . 7 (vol‘ℝ) = +∞
238232, 237eqtrdi 2789 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
239231, 238neeqtrrd 3015 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
240239necon2i 2975 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
241229, 240syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
242241expr 458 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
243 eqimss 4001 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
244243necon3bi 2967 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
245242, 244pm2.61d1 180 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2940  wral 3061  wrex 3070  Vcvv 3444  wss 3911  c0 4283  {csn 4587   cuni 4866   ciun 4955   class class class wbr 5106  cmpt 5189   Or wor 5545   × cxp 5632  dom cdm 5634  ran crn 5635  cima 5637  ccom 5638   Fn wfn 6492  wf 6493  ontowfo 6495  cfv 6497  (class class class)co 7358  cdom 8884  csdm 8885  Fincfn 8886  supcsup 9381  cr 11055  0cc0 11056  1c1 11057   + caddc 11059  +∞cpnf 11191  *cxr 11193   < clt 11194  cle 11195  cn 12158  cuz 12768  [,]cicc 13273  ...cfz 13430  Σcsu 15576  vol*covol 24842  volcvol 24843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-dju 9842  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-fl 13703  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577  df-rest 17309  df-topgen 17330  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-cmp 22754  df-ovol 24844  df-vol 24845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator