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Theorem voliunnfl 33768
Description: voliun 23541 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 4645 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4666 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2863 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6415 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 23526 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 23517 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 23480 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2835 . . . . . . 7 (vol‘∅) = 0
104, 9syl6req 2864 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8201 . . . . . . . . . . 11 Rel ≼
1312brrelexi 5365 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 8331 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 467 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 8353 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 578 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4670 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 612 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3259 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2220, 21bitr4i 269 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
23 ovolctb2 23479 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
2423ex 399 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
2524imdistani 560 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2625ralimi 3147 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2722, 26sylbi 208 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2827ancoms 448 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
29 foima 6339 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
3029raleqdv 3340 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
31 fofn 6336 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
32 ssid 3827 . . . . . . . . . . . 12 ℕ ⊆ ℕ
33 sseq1 3830 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → (𝑥 ⊆ ℝ ↔ (𝑔𝑚) ⊆ ℝ))
34 fveqeq2 6420 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑔𝑚)) = 0))
3533, 34anbi12d 618 . . . . . . . . . . . . 13 (𝑥 = (𝑔𝑚) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3635ralima 6726 . . . . . . . . . . . 12 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3731, 32, 36sylancl 576 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3830, 37bitr3d 272 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
39 difss 3943 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚)
40 ovolssnul 23474 . . . . . . . . . . . . . . . . . 18 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚) ∧ (𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
4139, 40mp3an1 1565 . . . . . . . . . . . . . . . . 17 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
42 ssdifss 3947 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ⊆ ℝ → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ)
43 nulmbl 23522 . . . . . . . . . . . . . . . . . . 19 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol)
44 mblvol 23517 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
4544eqeq1d 2815 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → ((vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 ↔ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0))
4645biimpar 465 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
47 0re 10330 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ
4846, 47syl6eqel 2900 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)
4948expcom 400 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5049ancld 542 . . . . . . . . . . . . . . . . . . . 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)))
5150adantl 469 . . . . . . . . . . . . . . . . . . 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)))
5243, 51mpd 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))
5342, 52sylan 571 . . . . . . . . . . . . . . . . 17 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5441, 53syldan 581 . . . . . . . . . . . . . . . 16 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5554ralimi 3147 . . . . . . . . . . . . . . 15 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ∀𝑚 ∈ ℕ (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
56 fveq2 6411 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
57 oveq2 6885 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1..^𝑚) = (1..^𝑛))
5857iuneq1d 4744 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 𝑙 ∈ (1..^𝑚)(𝑔𝑙) = 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
5956, 58difeq12d 3935 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
60 eqid 2813 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
61 fvex 6424 . . . . . . . . . . . . . . . . . . . . 21 (𝑔𝑛) ∈ V
62 difexg 5010 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑛) ∈ V → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V)
6361, 62ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V
6459, 60, 63fvmpt 6506 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
6564eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ↔ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol))
6664fveq2d 6415 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
6766eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ ↔ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ))
6865, 67anbi12d 618 . . . . . . . . . . . . . . . . 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)))
6968ralbiia 3174 . . . . . . . . . . . . . . . 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))
70 fveq2 6411 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
71 oveq2 6885 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7271iuneq1d 4744 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 𝑙 ∈ (1..^𝑛)(𝑔𝑙) = 𝑙 ∈ (1..^𝑚)(𝑔𝑙))
7370, 72difeq12d 3935 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
7473eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ↔ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol))
7573fveq2d 6415 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
7675eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → ((vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ ↔ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
7774, 76anbi12d 618 . . . . . . . . . . . . . . . . 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)))
7877cbvralv 3367 . . . . . . . . . . . . . . . 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))
7969, 78bitri 266 . . . . . . . . . . . . . . 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))
8055, 79sylibr 225 . . . . . . . . . . . . . 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))
81 fveq2 6411 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑙 → (𝑔𝑛) = (𝑔𝑙))
8281iundisj2 23536 . . . . . . . . . . . . . . 15 Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
83 disjeq2 4823 . . . . . . . . . . . . . . . 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))))
8483, 64mprg 3121 . . . . . . . . . . . . . . 15 (Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
8582, 84mpbir 222 . . . . . . . . . . . . . 14 Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)
86 nnex 11314 . . . . . . . . . . . . . . . . 17 ℕ ∈ V
8786mptex 6714 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ V
88 fveq1 6410 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
8988eleq1d 2877 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((𝑓𝑛) ∈ dom vol ↔ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol))
9088fveq2d 6415 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (vol‘(𝑓𝑛)) = (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
9190eleq1d 2877 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((vol‘(𝑓𝑛)) ∈ ℝ ↔ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ))
9289, 91anbi12d 618 . . . . . . . . . . . . . . . . . . 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)))
9392ralbidv 3181 . . . . . . . . . . . . . . . . . 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)))
9488adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9594disjeq2dv 4824 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (Disj 𝑛 ∈ ℕ (𝑓𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
9693, 95anbi12d 618 . . . . . . . . . . . . . . . . 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))))
9788iuneq2d 4746 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → 𝑛 ∈ ℕ (𝑓𝑛) = 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9897fveq2d 6415 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
99 voliunnfl.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑆 = seq1( + , 𝐺)
100 voliunnfl.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))
101 seqeq3 13032 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))))
102100, 101ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
10399, 102eqtri 2835 . . . . . . . . . . . . . . . . . . . . 21 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
104103rneqi 5560 . . . . . . . . . . . . . . . . . . . 20 ran 𝑆 = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
105104supeq1i 8595 . . . . . . . . . . . . . . . . . . 19 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))), ℝ*, < )
10690mpteq2dv 4946 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))))
107106seqeq3d 13035 . . . . . . . . . . . . . . . . . . . . 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)))))
108107rneqd 5561 . . . . . . . . . . . . . . . . . . . 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)))))
109108supeq1d 8594 . . . . . . . . . . . . . . . . . . 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*, < ))
110105, 109syl5eq 2859 . . . . . . . . . . . . . . . . . 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*, < ))
11198, 110eqeq12d 2828 . . . . . . . . . . . . . . . . 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*, < )))
11296, 111imbi12d 335 . . . . . . . . . . . . . . . 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*, < ))))
113 voliunnfl.3 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((𝑓𝑛) ∈ dom vol ∧ (vol‘(𝑓𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝑓𝑛)) = sup(ran 𝑆, ℝ*, < ))
11487, 112, 113vtocl 3459 . . . . . . . . . . . . . . 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*, < ))
11564iuneq2i 4738 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
116115fveq2i 6414 . . . . . . . . . . . . . . 15 (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
11766mpteq2ia 4941 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
118 seqeq3 13032 . . . . . . . . . . . . . . . . . 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))))))
119117, 118ax-mp 5 . . . . . . . . . . . . . . . . 17 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))))
120119rneqi 5560 . . . . . . . . . . . . . . . 16 ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))) = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))))
121120supeq1i 8595 . . . . . . . . . . . . . . 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*, < )
122114, 116, 1213eqtr3g 2870 . . . . . . . . . . . . . 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*, < ))
12380, 85, 122sylancl 576 . . . . . . . . . . . . 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*, < ))
124123adantl 469 . . . . . . . . . . . 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*, < ))
12581iundisj 23535 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ (𝑔𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
126 fofun 6335 . . . . . . . . . . . . . . . . 17 (𝑔:ℕ–onto𝐴 → Fun 𝑔)
127 funiunfv 6733 . . . . . . . . . . . . . . . . 17 (Fun 𝑔 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
128126, 127syl 17 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
129125, 128syl5eqr 2861 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = (𝑔 “ ℕ))
13029unieqd 4647 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 (𝑔 “ ℕ) = 𝐴)
131129, 130eqtrd 2847 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = 𝐴)
132131fveq2d 6415 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
133132adantr 468 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
13456sseq1d 3836 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((𝑔𝑚) ⊆ ℝ ↔ (𝑔𝑛) ⊆ ℝ))
13556fveqeq2d 6419 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((vol*‘(𝑔𝑚)) = 0 ↔ (vol*‘(𝑔𝑛)) = 0))
136134, 135anbi12d 618 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ↔ ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0)))
137136rspccva 3508 . . . . . . . . . . . . . . . . . . 19 ((∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ∧ 𝑛 ∈ ℕ) → ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0))
138 ssdifss 3947 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ⊆ ℝ → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
139138adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
140 difss 3943 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛)
141 ovolssnul 23474 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛) ∧ (𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
142140, 141mp3an1 1565 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
143139, 142jca 503 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0))
144 nulmbl 23522 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol)
145 mblvol 23517 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
146143, 144, 1453syl 18 . . . . . . . . . . . . . . . . . . . 20 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
147146, 142eqtrd 2847 . . . . . . . . . . . . . . . . . . 19 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
148137, 147syl 17 . . . . . . . . . . . . . . . . . 18 ((∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ∧ 𝑛 ∈ ℕ) → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
149148mpteq2dva 4945 . . . . . . . . . . . . . . . . 17 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))) = (𝑛 ∈ ℕ ↦ 0))
150149seqeq3d 13035 . . . . . . . . . . . . . . . 16 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = seq1( + , (𝑛 ∈ ℕ ↦ 0)))
151150rneqd 5561 . . . . . . . . . . . . . . 15 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = ran seq1( + , (𝑛 ∈ ℕ ↦ 0)))
152151supeq1d 8594 . . . . . . . . . . . . . 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*, < ))
153 0cn 10320 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℂ
154 ser1const 13083 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℂ ∧ 𝑚 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
155153, 154mpan 673 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
156 nncn 11316 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
157156mul01d 10523 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (𝑚 · 0) = 0)
158155, 157eqtrd 2847 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = 0)
159158mpteq2ia 4941 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)) = (𝑚 ∈ ℕ ↦ 0)
160 fconstmpt 5370 . . . . . . . . . . . . . . . . . . . . 21 (ℕ × {0}) = (𝑛 ∈ ℕ ↦ 0)
161 seqeq3 13032 . . . . . . . . . . . . . . . . . . . . 21 ((ℕ × {0}) = (𝑛 ∈ ℕ ↦ 0) → seq1( + , (ℕ × {0})) = seq1( + , (𝑛 ∈ ℕ ↦ 0)))
162160, 161ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (ℕ × {0})) = seq1( + , (𝑛 ∈ ℕ ↦ 0))
163 1z 11676 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
164 seqfn 13039 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
165163, 164ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
166 nnuz 11944 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
167166fneq2i 6200 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
168 dffn5 6465 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
169167, 168bitr3i 268 . . . . . . . . . . . . . . . . . . . . 21 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
170165, 169mpbi 221 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
171162, 170eqtr3i 2837 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
172 fconstmpt 5370 . . . . . . . . . . . . . . . . . . 19 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
173159, 171, 1723eqtr4i 2845 . . . . . . . . . . . . . . . . . 18 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (ℕ × {0})
174173rneqi 5560 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
175 1nn 11319 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
176 ne0i 4129 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℕ → ℕ ≠ ∅)
177 rnxp 5782 . . . . . . . . . . . . . . . . . 18 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
178175, 176, 177mp2b 10 . . . . . . . . . . . . . . . . 17 ran (ℕ × {0}) = {0}
179174, 178eqtri 2835 . . . . . . . . . . . . . . . 16 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = {0}
180179supeq1i 8595 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
181 xrltso 12193 . . . . . . . . . . . . . . . 16 < Or ℝ*
182 0xr 10374 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
183 supsn 8620 . . . . . . . . . . . . . . . 16 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
184181, 182, 183mp2an 675 . . . . . . . . . . . . . . 15 sup({0}, ℝ*, < ) = 0
185180, 184eqtri 2835 . . . . . . . . . . . . . 14 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
186152, 185syl6eq 2863 . . . . . . . . . . . . 13 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = 0)
187186adantl 469 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = 0)
188124, 133, 1873eqtr3rd 2856 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → 0 = (vol‘ 𝐴))
189188ex 399 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → 0 = (vol‘ 𝐴)))
19038, 189sylbid 231 . . . . . . . . 9 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → 0 = (vol‘ 𝐴)))
19128, 190syl5 34 . . . . . . . 8 (𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
192191exlimiv 2021 . . . . . . 7 (∃𝑔 𝑔:ℕ–onto𝐴 → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
19318, 192syl 17 . . . . . 6 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → 0 = (vol‘ 𝐴)))
194193expimpd 443 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
19511, 194pm2.61ine 3068 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
196 renepnf 10375 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
19747, 196mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
198 fveq2 6411 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
199 rembl 23527 . . . . . . . . 9 ℝ ∈ dom vol
200 mblvol 23517 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
201199, 200ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
202 ovolre 23512 . . . . . . . 8 (vol*‘ℝ) = +∞
203201, 202eqtri 2835 . . . . . . 7 (vol‘ℝ) = +∞
204198, 203syl6eq 2863 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
205197, 204neeqtrrd 3059 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
206205necon2i 3019 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
207195, 206syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
208207expr 446 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
209 eqimss 3861 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
210209necon3bi 3011 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
211208, 210pm2.61d1 172 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2157  wne 2985  wral 3103  Vcvv 3398  cdif 3773  wss 3776  c0 4123  {csn 4377   cuni 4637   ciun 4719  Disj wdisj 4819   class class class wbr 4851  cmpt 4930   Or wor 5238   × cxp 5316  dom cdm 5318  ran crn 5319  cima 5321  Fun wfun 6098   Fn wfn 6099  ontowfo 6102  cfv 6104  (class class class)co 6877  cdom 8193  csdm 8194  supcsup 8588  cc 10222  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   · cmul 10229  +∞cpnf 10359  *cxr 10361   < clt 10362  cn 11308  cz 11646  cuz 11907  ..^cfzo 12692  seqcseq 13027  vol*covol 23449  volcvol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-disj 4820  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-of 7130  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-cda 9278  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-n0 11563  df-z 11647  df-uz 11908  df-q 12011  df-rp 12050  df-xneg 12165  df-xadd 12166  df-xmul 12167  df-ioo 12400  df-ico 12402  df-icc 12403  df-fz 12553  df-fzo 12693  df-fl 12820  df-seq 13028  df-exp 13087  df-hash 13341  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-clim 14445  df-sum 14643  df-rest 16291  df-topgen 16312  df-psmet 19949  df-xmet 19950  df-met 19951  df-bl 19952  df-mopn 19953  df-top 20916  df-topon 20933  df-bases 20968  df-cmp 21408  df-ovol 23451  df-vol 23452
This theorem is referenced by: (None)
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