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 37269
Description: volsup 25529 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 4920 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4939 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2781 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6900 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 25512 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 25503 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 25466 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2753 . . . . . . 7 (vol‘∅) = 0
104, 9eqtr2di 2782 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8970 . . . . . . . . . . 11 Rel ≼
1312brrelex1i 5734 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 9129 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 478 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 9153 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 586 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4943 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 622 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3100 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 277 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 25465 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
24 nulmbl 25508 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 𝑥 ∈ dom vol)
25 mblvol 25503 . . . . . . . . . . . . . . . 16 (𝑥 ∈ dom vol → (vol‘𝑥) = (vol*‘𝑥))
26 eqtr 2748 . . . . . . . . . . . . . . . . 17 (((vol‘𝑥) = (vol*‘𝑥) ∧ (vol*‘𝑥) = 0) → (vol‘𝑥) = 0)
2726expcom 412 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol → (vol‘𝑥) = 0))
3024, 29jcai 515 . . . . . . . . . . . . 13 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3123, 30syldan 589 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
3231ralimi 3072 . . . . . . . . . . 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 457 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0))
35 fzfi 13973 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ Fin
36 fzssuz 13577 . . . . . . . . . . . . . . . . 17 (1...𝑚) ⊆ (ℤ‘1)
37 nnuz 12898 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
3836, 37sseqtrri 4014 . . . . . . . . . . . . . . . 16 (1...𝑚) ⊆ ℕ
39 fof 6810 . . . . . . . . . . . . . . . . . . . 20 (𝑔:ℕ–onto𝐴𝑔:ℕ⟶𝐴)
4039ffvelcdmda 7093 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) → (𝑔𝑙) ∈ 𝐴)
41 eleq1 2813 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → (𝑥 ∈ dom vol ↔ (𝑔𝑙) ∈ dom vol))
42 fveqeq2 6905 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑔𝑙) → ((vol‘𝑥) = 0 ↔ (vol‘(𝑔𝑙)) = 0))
4341, 42anbi12d 630 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑔𝑙) → ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ↔ ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0)))
4443rspccva 3605 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → ((𝑔𝑙) ∈ dom vol ∧ (vol‘(𝑔𝑙)) = 0))
4544simpld 493 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (𝑔𝑙) ∈ dom vol)
4645ancoms 457 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑙) ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4740, 46sylan 578 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑔𝑙) ∈ dom vol)
4847an32s 650 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (𝑔𝑙) ∈ dom vol)
4948ralrimiva 3135 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (𝑔𝑙) ∈ dom vol)
50 ssralv 4045 . . . . . . . . . . . . . . . 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 25517 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ ∀𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5335, 51, 52sylancr 585 . . . . . . . . . . . . . 14 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5453adantr 479 . . . . . . . . . . . . 13 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ dom vol)
5554fmpttd 7124 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol)
56 fzssp1 13579 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ (1...(𝑛 + 1))
57 iunss1 5011 . . . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
6059iuneq1d 5024 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
61 eqid 2725 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
62 ovex 7452 . . . . . . . . . . . . . . . . 17 (1...𝑛) ∈ V
63 fvex 6909 . . . . . . . . . . . . . . . . 17 (𝑔𝑙) ∈ V
6462, 63iunex 7973 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑛)(𝑔𝑙) ∈ V
6560, 61, 64fvmpt 7004 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) = 𝑙 ∈ (1...𝑛)(𝑔𝑙))
66 peano2nn 12257 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
67 oveq2 7427 . . . . . . . . . . . . . . . . . 18 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
6867iuneq1d 5024 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 + 1) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) = 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙))
69 ovex 7452 . . . . . . . . . . . . . . . . . 18 (1...(𝑛 + 1)) ∈ V
7069, 63iunex 7973 . . . . . . . . . . . . . . . . 17 𝑙 ∈ (1...(𝑛 + 1))(𝑔𝑙) ∈ V
7168, 61, 70fvmpt 7004 . . . . . . . . . . . . . . . 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 4010 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
7574rgen 3052 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1))
76 nnex 12251 . . . . . . . . . . . . . 14 ℕ ∈ V
7776mptex 7235 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ V
78 feq1 6704 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓:ℕ⟶dom vol ↔ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol))
79 fveq1 6895 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘𝑛))
80 fveq1 6895 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (𝑓‘(𝑛 + 1)) = ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))‘(𝑛 + 1)))
8179, 80sseq12d 4010 . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 5938 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8584unieqd 4922 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → ran 𝑓 = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
8685fveq2d 6900 . . . . . . . . . . . . . . 15 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol‘ ran 𝑓) = (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8784imaeq2d 6064 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) → (vol “ ran 𝑓) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))))
8887supeq1d 9471 . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . 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 3536 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 4999 . . . . . . . . . . . . . . . 16 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)}
95 eluzfz2 13544 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ (1...𝑥))
9695, 37eleq2s 2843 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℕ → 𝑥 ∈ (1...𝑥))
97 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑥 → (𝑔𝑙) = (𝑔𝑥))
9897eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑥 → (𝑛 ∈ (𝑔𝑙) ↔ 𝑛 ∈ (𝑔𝑥)))
9998rspcev 3606 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (1...𝑥) ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
10096, 99sylan 578 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙))
101 oveq2 7427 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑥 → (1...𝑚) = (1...𝑥))
102101rexeqdv 3315 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑥 → (∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)))
103102rspcev 3606 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℕ ∧ ∃𝑙 ∈ (1...𝑥)𝑛 ∈ (𝑔𝑙)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
104100, 103syldan 589 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑛 ∈ (𝑔𝑥)) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
105104rexlimiva 3136 . . . . . . . . . . . . . . . . . . 19 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) → ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
106 ssrexv 4046 . . . . . . . . . . . . . . . . . . . . . 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 3140 . . . . . . . . . . . . . . . . . . 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 5001 . . . . . . . . . . . . . . . . . . 19 (𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
113112rexbii 3083 . . . . . . . . . . . . . . . . . 18 (∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙) ↔ ∃𝑚 ∈ ℕ ∃𝑙 ∈ (1...𝑚)𝑛 ∈ (𝑔𝑙))
114111, 113bitr4i 277 . . . . . . . . . . . . . . . . 17 (∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥) ↔ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙))
115114abbii 2795 . . . . . . . . . . . . . . . 16 {𝑛 ∣ ∃𝑥 ∈ ℕ 𝑛 ∈ (𝑔𝑥)} = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
11694, 115eqtri 2753 . . . . . . . . . . . . . . 15 𝑥 ∈ ℕ (𝑔𝑥) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
117 df-iun 4999 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = {𝑛 ∣ ∃𝑚 ∈ ℕ 𝑛 𝑙 ∈ (1...𝑚)(𝑔𝑙)}
118 ovex 7452 . . . . . . . . . . . . . . . . 17 (1...𝑚) ∈ V
119118, 63iunex 7973 . . . . . . . . . . . . . . . 16 𝑙 ∈ (1...𝑚)(𝑔𝑙) ∈ V
120119dfiun3 5969 . . . . . . . . . . . . . . 15 𝑚 ∈ ℕ 𝑙 ∈ (1...𝑚)(𝑔𝑙) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
121116, 117, 1203eqtr2i 2759 . . . . . . . . . . . . . 14 𝑥 ∈ ℕ (𝑔𝑥) = ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))
122 fofn 6812 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
123 fniunfv 7257 . . . . . . . . . . . . . . . 16 (𝑔 Fn ℕ → 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
124122, 123syl 17 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = ran 𝑔)
125 forn 6813 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 → ran 𝑔 = 𝐴)
126125unieqd 4922 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 ran 𝑔 = 𝐴)
127124, 126eqtrd 2765 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑥 ∈ ℕ (𝑔𝑥) = 𝐴)
128121, 127eqtr3id 2779 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 𝐴)
129128fveq2d 6900 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
130129adantr 479 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol‘ 𝐴))
131 rnco2 6259 . . . . . . . . . . . . . 14 ran (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
132 eqidd 2726 . . . . . . . . . . . . . . . . 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 25502 . . . . . . . . . . . . . . . . . . 19 vol:dom vol⟶(0[,]+∞)
134133a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol:dom vol⟶(0[,]+∞))
135134feqmptd 6966 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → vol = (𝑛 ∈ dom vol ↦ (vol‘𝑛)))
136 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑙 ∈ (1...𝑚)(𝑔𝑙) → (vol‘𝑛) = (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)))
13754, 132, 135, 136fmptco 7138 . . . . . . . . . . . . . . . 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 25503 . . . . . . . . . . . . . . . . . . . 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 25504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
141140adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → 𝑥 ⊆ ℝ)
14225eqeq1d 2727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 ↔ (vol*‘𝑥) = 0))
143 0re 11248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ ℝ
144 eleq1a 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 ∈ ℝ → ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
145143, 144ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((vol*‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ)
146142, 145biimtrdi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ dom vol → ((vol‘𝑥) = 0 → (vol*‘𝑥) ∈ ℝ))
147146imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (vol*‘𝑥) ∈ ℝ)
148141, 147jca 510 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
149148ralimi 3072 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
150149adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ))
151 ssid 3999 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℕ ⊆ ℕ
152 sseq1 4002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → (𝑥 ⊆ ℝ ↔ (𝑔𝑙) ⊆ ℝ))
153 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑔𝑙) → (vol*‘𝑥) = (vol*‘(𝑔𝑙)))
154153eleq1d 2810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑔𝑙) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘(𝑔𝑙)) ∈ ℝ))
155152, 154anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝑔𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
156155ralima 7250 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
157122, 151, 156sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ)))
158 foima 6815 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
159158raleqdv 3314 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
160157, 159bitr3d 280 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
161160adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ ((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
163 ssralv 4045 . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)((𝑔𝑙) ⊆ ℝ ∧ (vol*‘(𝑔𝑙)) ∈ ℝ))
166 ovolfiniun 25474 . . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . . 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 25503 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔𝑙) ∈ 𝐴) → (vol‘(𝑔𝑙)) = 0)
17140, 170sylan2 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0) ∧ (𝑔:ℕ–onto𝐴𝑙 ∈ ℕ)) → (vol‘(𝑔𝑙)) = 0)
172171ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑔:ℕ–onto𝐴𝑙 ∈ ℕ) ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol‘(𝑔𝑙)) = 0)
173172an32s 650 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol‘(𝑔𝑙)) = 0)
174169, 173eqtr3d 2767 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑙 ∈ ℕ) → (vol*‘(𝑔𝑙)) = 0)
175174ralrimiva 3135 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → ∀𝑙 ∈ ℕ (vol*‘(𝑔𝑙)) = 0)
176 ssralv 4045 . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → ∀𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
179178sumeq2d 15684 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = Σ𝑙 ∈ (1...𝑚)0)
18035olci 864 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin)
181 sumz 15704 . . . . . . . . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → Σ𝑙 ∈ (1...𝑚)(vol*‘(𝑔𝑙)) = 0)
184167, 183breqtrd 5175 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ≤ 0)
185 mblss 25504 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑙) ∈ dom vol → (𝑔𝑙) ⊆ ℝ)
186185ralimi 3072 . . . . . . . . . . . . . . . . . . . . . . . 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 5049 . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → 𝑙 ∈ (1...𝑚)(𝑔𝑙) ⊆ ℝ)
191 ovolge0 25454 . . . . . . . . . . . . . . . . . . . . 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 25451 . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ∈ ℝ*)
196 0xr 11293 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ*
197 xrletri3 13168 . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol*‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
200139, 199eqtrd 2765 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) ∧ 𝑚 ∈ ℕ) → (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) = 0)
201200mpteq2dva 5249 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ 0))
202 fconstmpt 5740 . . . . . . . . . . . . . . . . 17 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
203201, 202eqtr4di 2783 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (𝑚 ∈ ℕ ↦ (vol‘ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
204137, 203eqtrd 2765 . . . . . . . . . . . . . . 15 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol ∘ (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = (ℕ × {0}))
205 frn 6730 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)):ℕ⟶dom vol → ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) ⊆ dom vol)
206 ffn 6723 . . . . . . . . . . . . . . . . . . 19 (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
207133, 206ax-mp 5 . . . . . . . . . . . . . . . . . 18 vol Fn dom vol
208119, 61fnmpti 6699 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙)) Fn ℕ
209 fnco 6673 . . . . . . . . . . . . . . . . . 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 1447 . . . . . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ
213212ne0ii 4337 . . . . . . . . . . . . . . . 16 ℕ ≠ ∅
214 fconst5 7218 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . 13 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → (vol “ ran (𝑚 ∈ ℕ ↦ 𝑙 ∈ (1...𝑚)(𝑔𝑙))) = {0})
218217supeq1d 9471 . . . . . . . . . . . 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 13155 . . . . . . . . . . . . 13 < Or ℝ*
220 supsn 9497 . . . . . . . . . . . . 13 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
221219, 196, 220mp2an 690 . . . . . . . . . . . 12 sup({0}, ℝ*, < ) = 0
222218, 221eqtrdi 2781 . . . . . . . . . . 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 2774 . . . . . . . . . 10 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ∈ dom vol ∧ (vol‘𝑥) = 0)) → 0 = (vol‘ 𝐴))
224223ex 411 . . . . . . . . 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 1925 . . . . . . 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 452 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
22911, 228pm2.61ine 3014 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
230 renepnf 11294 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
231143, 230mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
232 fveq2 6896 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
233 rembl 25513 . . . . . . . . 9 ℝ ∈ dom vol
234 mblvol 25503 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
235233, 234ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
236 ovolre 25498 . . . . . . . 8 (vol*‘ℝ) = +∞
237235, 236eqtri 2753 . . . . . . 7 (vol‘ℝ) = +∞
238232, 237eqtrdi 2781 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
239231, 238neeqtrrd 3004 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
240239necon2i 2964 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
241229, 240syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
242241expr 455 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
243 eqimss 4035 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
244243necon3bi 2956 . 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 394  wo 845   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wne 2929  wral 3050  wrex 3059  Vcvv 3461  wss 3944  c0 4322  {csn 4630   cuni 4909   ciun 4997   class class class wbr 5149  cmpt 5232   Or wor 5589   × cxp 5676  dom cdm 5678  ran crn 5679  cima 5681  ccom 5682   Fn wfn 6544  wf 6545  ontowfo 6547  cfv 6549  (class class class)co 7419  cdom 8962  csdm 8963  Fincfn 8964  supcsup 9465  cr 11139  0cc0 11140  1c1 11141   + caddc 11143  +∞cpnf 11277  *cxr 11279   < clt 11280  cle 11281  cn 12245  cuz 12855  [,]cicc 13362  ...cfz 13519  Σcsu 15668  vol*covol 25435  volcvol 25436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-rest 17407  df-topgen 17428  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-top 22840  df-topon 22857  df-bases 22893  df-cmp 23335  df-ovol 25437  df-vol 25438
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator