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

Theorem voliunnfl 33583
Description: voliun 23368 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
Hypotheses
Ref Expression
voliunnfl.1 𝑆 = seq1( + , 𝐺)
voliunnfl.2 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))
voliunnfl.3 ((∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = sup(ran 𝑆, ℝ*, < ))
Assertion
Ref Expression
voliunnfl ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Distinct variable group:   𝑓,𝑛,𝑥,𝐴
Allowed substitution hints:   𝑆(𝑥,𝑓,𝑛)   𝐺(𝑥,𝑓,𝑛)

Proof of Theorem voliunnfl
Dummy variables 𝑔 𝑚 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4476 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4497 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2701 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6233 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 23353 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 23344 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 23307 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2673 . . . . . . 7 (vol‘∅) = 0
104, 9syl6req 2702 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8003 . . . . . . . . . . 11 Rel ≼
1312brrelexi 5192 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 8130 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 503 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 8152 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 702 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4501 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 731 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3093 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 267 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 23306 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
2423ex 449 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
2524imdistani 726 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2625ralimi 2981 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2722, 26sylbi 207 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2827ancoms 468 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
29 foima 6158 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
3029raleqdv 3174 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
31 fofn 6155 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
32 ssid 3657 . . . . . . . . . . . 12 ℕ ⊆ ℕ
33 sseq1 3659 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → (𝑥 ⊆ ℝ ↔ (𝑔𝑚) ⊆ ℝ))
34 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = (𝑔𝑚) → (vol*‘𝑥) = (vol*‘(𝑔𝑚)))
3534eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑔𝑚)) = 0))
3633, 35anbi12d 747 . . . . . . . . . . . . 13 (𝑥 = (𝑔𝑚) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3736ralima 6538 . . . . . . . . . . . 12 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3831, 32, 37sylancl 695 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3930, 38bitr3d 270 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
40 difss 3770 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚)
41 ovolssnul 23301 . . . . . . . . . . . . . . . . . 18 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚) ∧ (𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
4240, 41mp3an1 1451 . . . . . . . . . . . . . . . . 17 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
43 ssdifss 3774 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ⊆ ℝ → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ)
44 nulmbl 23349 . . . . . . . . . . . . . . . . . . 19 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol)
45 mblvol 23344 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
4645eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → ((vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 ↔ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0))
4746biimpar 501 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
48 0re 10078 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ
4947, 48syl6eqel 2738 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)
5049expcom 450 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5150ancld 575 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)))
5251adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)))
5344, 52mpd 15 . . . . . . . . . . . . . . . . . 18 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5443, 53sylan 487 . . . . . . . . . . . . . . . . 17 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5542, 54syldan 486 . . . . . . . . . . . . . . . 16 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5655ralimi 2981 . . . . . . . . . . . . . . 15 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ∀𝑚 ∈ ℕ (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
57 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
58 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1..^𝑚) = (1..^𝑛))
5958iuneq1d 4577 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 𝑙 ∈ (1..^𝑚)(𝑔𝑙) = 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
6057, 59difeq12d 3762 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
61 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
62 fvex 6239 . . . . . . . . . . . . . . . . . . . . 21 (𝑔𝑛) ∈ V
63 difexg 4841 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑛) ∈ V → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V)
6462, 63ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V
6560, 61, 64fvmpt 6321 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
6665eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ↔ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol))
6765fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
6867eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ ↔ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ))
6966, 68anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ↔ (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ)))
7069ralbiia 3008 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ↔ ∀𝑛 ∈ ℕ (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ))
71 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
72 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7372iuneq1d 4577 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 𝑙 ∈ (1..^𝑛)(𝑔𝑙) = 𝑙 ∈ (1..^𝑚)(𝑔𝑙))
7471, 73difeq12d 3762 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
7574eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ↔ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol))
7674fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
7776eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → ((vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ ↔ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
7875, 77anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ) ↔ (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)))
7978cbvralv 3201 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ ℕ (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ) ↔ ∀𝑚 ∈ ℕ (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
8070, 79bitri 264 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ↔ ∀𝑚 ∈ ℕ (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
8156, 80sylibr 224 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ))
82 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑙 → (𝑔𝑛) = (𝑔𝑙))
8382iundisj2 23363 . . . . . . . . . . . . . . 15 Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
84 disjeq2 4656 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) → (Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
8584, 65mprg 2955 . . . . . . . . . . . . . . 15 (Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
8683, 85mpbir 221 . . . . . . . . . . . . . 14 Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)
87 nnex 11064 . . . . . . . . . . . . . . . . 17 ℕ ∈ V
8887mptex 6527 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ V
89 fveq1 6228 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9089eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((𝑓𝑛) ∈ dom vol ↔ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol))
9189fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (vol‘(𝑓𝑛)) = (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
9291eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((vol‘(𝑓𝑛)) ∈ ℝ ↔ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ))
9390, 92anbi12d 747 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ↔ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ)))
9493ralbidv 3015 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ↔ ∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ)))
9589adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9695disjeq2dv 4657 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (Disj 𝑛 ∈ ℕ (𝑓𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
9794, 96anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)) ↔ (∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))))
9889iuneq2d 4579 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → 𝑛 ∈ ℕ (𝑓𝑛) = 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9998fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
100 voliunnfl.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑆 = seq1( + , 𝐺)
101 voliunnfl.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))
102 seqeq3 12846 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))))
103101, 102ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
104100, 103eqtri 2673 . . . . . . . . . . . . . . . . . . . . 21 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
105104rneqi 5384 . . . . . . . . . . . . . . . . . . . 20 ran 𝑆 = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
106105supeq1i 8394 . . . . . . . . . . . . . . . . . . 19 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))), ℝ*, < )
10791mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))))
108107seqeq3d 12849 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))))
109108rneqd 5385 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))) = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))))
110109supeq1d 8393 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))), ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < ))
111106, 110syl5eq 2697 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < ))
11299, 111eqeq12d 2666 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = sup(ran 𝑆, ℝ*, < ) ↔ (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < )))
11397, 112imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (((∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = sup(ran 𝑆, ℝ*, < )) ↔ ((∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) → (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < ))))
114 voliunnfl.3 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = sup(ran 𝑆, ℝ*, < ))
11588, 113, 114vtocl 3290 . . . . . . . . . . . . . . 15 ((∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) → (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < ))
11665iuneq2i 4571 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
117116fveq2i 6232 . . . . . . . . . . . . . . 15 (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
11867mpteq2ia 4773 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
119 seqeq3 12846 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))) → seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))))
120118, 119ax-mp 5 . . . . . . . . . . . . . . . . 17 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))))
121120rneqi 5384 . . . . . . . . . . . . . . . 16 ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))) = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))))
122121supeq1i 8394 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))), ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < )
123115, 117, 1223eqtr3g 2708 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ ℕ (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ∧ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ))
12481, 86, 123sylancl 695 . . . . . . . . . . . . 13 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ))
125124adantl 481 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ))
12682iundisj 23362 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ (𝑔𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
127 fofun 6154 . . . . . . . . . . . . . . . . 17 (𝑔:ℕ–onto𝐴 → Fun 𝑔)
128 funiunfv 6546 . . . . . . . . . . . . . . . . 17 (Fun 𝑔 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
129127, 128syl 17 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
130126, 129syl5eqr 2699 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = (𝑔 “ ℕ))
13129unieqd 4478 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 (𝑔 “ ℕ) = 𝐴)
132130, 131eqtrd 2685 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = 𝐴)
133132fveq2d 6233 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
134133adantr 480 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
13557sseq1d 3665 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((𝑔𝑚) ⊆ ℝ ↔ (𝑔𝑛) ⊆ ℝ))
13657fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (vol*‘(𝑔𝑚)) = (vol*‘(𝑔𝑛)))
137136eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((vol*‘(𝑔𝑚)) = 0 ↔ (vol*‘(𝑔𝑛)) = 0))
138135, 137anbi12d 747 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ↔ ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0)))
139138rspccva 3339 . . . . . . . . . . . . . . . . . . 19 ((∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ∧ 𝑛 ∈ ℕ) → ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0))
140 ssdifss 3774 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ⊆ ℝ → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
141140adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
142 difss 3770 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛)
143 ovolssnul 23301 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛) ∧ (𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
144142, 143mp3an1 1451 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
145141, 144jca 553 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0))
146 nulmbl 23349 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol)
147 mblvol 23344 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
148145, 146, 1473syl 18 . . . . . . . . . . . . . . . . . . . 20 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
149148, 144eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
150139, 149syl 17 . . . . . . . . . . . . . . . . . 18 ((∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ∧ 𝑛 ∈ ℕ) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
151150mpteq2dva 4777 . . . . . . . . . . . . . . . . 17 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))) = (𝑛 ∈ ℕ ↦ 0))
152151seqeq3d 12849 . . . . . . . . . . . . . . . 16 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = seq1( + , (𝑛 ∈ ℕ ↦ 0)))
153152rneqd 5385 . . . . . . . . . . . . . . 15 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = ran seq1( + , (𝑛 ∈ ℕ ↦ 0)))
154153supeq1d 8393 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ))
155 0cn 10070 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℂ
156 ser1const 12897 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℂ ∧ 𝑚 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
157155, 156mpan 706 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
158 nncn 11066 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
159158mul01d 10273 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (𝑚 · 0) = 0)
160157, 159eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = 0)
161160mpteq2ia 4773 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)) = (𝑚 ∈ ℕ ↦ 0)
162 fconstmpt 5197 . . . . . . . . . . . . . . . . . . . . 21 (ℕ × {0}) = (𝑛 ∈ ℕ ↦ 0)
163 seqeq3 12846 . . . . . . . . . . . . . . . . . . . . 21 ((ℕ × {0}) = (𝑛 ∈ ℕ ↦ 0) → seq1( + , (ℕ × {0})) = seq1( + , (𝑛 ∈ ℕ ↦ 0)))
164162, 163ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (ℕ × {0})) = seq1( + , (𝑛 ∈ ℕ ↦ 0))
165 1z 11445 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
166 seqfn 12853 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
167165, 166ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
168 nnuz 11761 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
169168fneq2i 6024 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
170 dffn5 6280 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
171169, 170bitr3i 266 . . . . . . . . . . . . . . . . . . . . 21 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
172167, 171mpbi 220 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
173164, 172eqtr3i 2675 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
174 fconstmpt 5197 . . . . . . . . . . . . . . . . . . 19 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
175161, 173, 1743eqtr4i 2683 . . . . . . . . . . . . . . . . . 18 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (ℕ × {0})
176175rneqi 5384 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
177 1nn 11069 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
178 ne0i 3954 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℕ → ℕ ≠ ∅)
179 rnxp 5599 . . . . . . . . . . . . . . . . . 18 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
180177, 178, 179mp2b 10 . . . . . . . . . . . . . . . . 17 ran (ℕ × {0}) = {0}
181176, 180eqtri 2673 . . . . . . . . . . . . . . . 16 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = {0}
182181supeq1i 8394 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
183 xrltso 12012 . . . . . . . . . . . . . . . 16 < Or ℝ*
184 0xr 10124 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
185 supsn 8419 . . . . . . . . . . . . . . . 16 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
186183, 184, 185mp2an 708 . . . . . . . . . . . . . . 15 sup({0}, ℝ*, < ) = 0
187182, 186eqtri 2673 . . . . . . . . . . . . . 14 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
188154, 187syl6eq 2701 . . . . . . . . . . . . 13 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = 0)
189188adantl 481 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = 0)
190125, 134, 1893eqtr3rd 2694 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → 0 = (vol‘ 𝐴))
191190ex 449 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → 0 = (vol‘ 𝐴)))
19239, 191sylbid 230 . . . . . . . . 9 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 0 = (vol‘ 𝐴)))
19328, 192syl5 34 . . . . . . . 8 (𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
194193exlimiv 1898 . . . . . . 7 (∃𝑔 𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
19518, 194syl 17 . . . . . 6 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
196195expimpd 628 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
19711, 196pm2.61ine 2906 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
198 renepnf 10125 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
19948, 198mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
200 fveq2 6229 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
201 rembl 23354 . . . . . . . . 9 ℝ ∈ dom vol
202 mblvol 23344 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
203201, 202ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
204 ovolre 23339 . . . . . . . 8 (vol*‘ℝ) = +∞
205203, 204eqtri 2673 . . . . . . 7 (vol‘ℝ) = +∞
206200, 205syl6eq 2701 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
207199, 206neeqtrrd 2897 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
208207necon2i 2857 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
209197, 208syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
210209expr 642 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
211 eqimss 3690 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
212211necon3bi 2849 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
213210, 212pm2.61d1 171 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210   cuni 4468   ciun 4552  Disj wdisj 4652   class class class wbr 4685  cmpt 4762   Or wor 5063   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  ontowfo 5924  cfv 5926  (class class class)co 6690  cdom 7995  csdm 7996  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cn 11058  cz 11415  cuz 11725  ..^cfzo 12504  seqcseq 12841  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator