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Theorem voliunnfl 37378
Description: voliun 25571 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 4916 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4935 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2782 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6897 . . . . . . 7 (𝐴 = ∅ → (vol‘ 𝐴) = (vol‘∅))
5 0mbl 25556 . . . . . . . . 9 ∅ ∈ dom vol
6 mblvol 25547 . . . . . . . . 9 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
75, 6ax-mp 5 . . . . . . . 8 (vol‘∅) = (vol*‘∅)
8 ovol0 25510 . . . . . . . 8 (vol*‘∅) = 0
97, 8eqtri 2754 . . . . . . 7 (vol‘∅) = 0
104, 9eqtr2di 2783 . . . . . 6 (𝐴 = ∅ → 0 = (vol‘ 𝐴))
1110a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
12 reldom 8972 . . . . . . . . . . 11 Rel ≼
1312brrelex1i 5730 . . . . . . . . . 10 (𝐴 ≼ ℕ → 𝐴 ∈ V)
14 0sdomg 9134 . . . . . . . . . 10 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1513, 14syl 17 . . . . . . . . 9 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1615biimparc 478 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
17 fodomr 9158 . . . . . . . 8 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
1816, 17sylancom 586 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto𝐴)
19 unissb 4939 . . . . . . . . . . . . 13 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
2019anbi1i 622 . . . . . . . . . . . 12 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
21 r19.26 3101 . . . . . . . . . . . 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 25509 . . . . . . . . . . . . . 14 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (vol*‘𝑥) = 0)
2423ex 411 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
2524imdistani 567 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2625ralimi 3073 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2722, 26sylbi 216 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
2827ancoms 457 . . . . . . . . 9 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
29 foima 6812 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴 → (𝑔 “ ℕ) = 𝐴)
3029raleqdv 3315 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
31 fofn 6809 . . . . . . . . . . . 12 (𝑔:ℕ–onto𝐴𝑔 Fn ℕ)
32 ssid 4001 . . . . . . . . . . . 12 ℕ ⊆ ℕ
33 sseq1 4004 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → (𝑥 ⊆ ℝ ↔ (𝑔𝑚) ⊆ ℝ))
34 fveqeq2 6902 . . . . . . . . . . . . . 14 (𝑥 = (𝑔𝑚) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑔𝑚)) = 0))
3533, 34anbi12d 630 . . . . . . . . . . . . 13 (𝑥 = (𝑔𝑚) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3635ralima 7247 . . . . . . . . . . . 12 ((𝑔 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3731, 32, 36sylancl 584 . . . . . . . . . . 11 (𝑔:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑔 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
3830, 37bitr3d 280 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)))
39 difss 4128 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚)
40 ovolssnul 25504 . . . . . . . . . . . . . . . . . 18 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ (𝑔𝑚) ∧ (𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
4139, 40mp3an1 1445 . . . . . . . . . . . . . . . . 17 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
42 ssdifss 4132 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑚) ⊆ ℝ → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ)
43 nulmbl 25552 . . . . . . . . . . . . . . . . . . 19 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol)
44 mblvol 25547 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
4544eqeq1d 2728 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → ((vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 ↔ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0))
4645biimpar 476 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0)
47 0re 11257 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ
4846, 47eqeltrdi 2834 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0) → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ)
4948expcom 412 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = 0 → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol → (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5049ancld 549 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . . . 16 (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol ∧ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
5554ralimi 3073 . . . . . . . . . . . . . . 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 6893 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
57 oveq2 7424 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1..^𝑚) = (1..^𝑛))
5857iuneq1d 5020 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 𝑙 ∈ (1..^𝑚)(𝑔𝑙) = 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
5956, 58difeq12d 4119 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
60 eqid 2726 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
61 fvex 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑔𝑛) ∈ V
62 difexg 5326 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔𝑛) ∈ V → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V)
6361, 62ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ V
6459, 60, 63fvmpt 7001 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
6564eleq1d 2811 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol ↔ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol))
6664fveq2d 6897 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
6766eleq1d 2811 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ ↔ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ))
6865, 67anbi12d 630 . . . . . . . . . . . . . . . . 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 3081 . . . . . . . . . . . . . . . 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 6893 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
71 oveq2 7424 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7271iuneq1d 5020 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑚 𝑙 ∈ (1..^𝑛)(𝑔𝑙) = 𝑙 ∈ (1..^𝑚)(𝑔𝑙))
7370, 72difeq12d 4119 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))
7473eleq1d 2811 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol ↔ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)) ∈ dom vol))
7573fveq2d 6897 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))))
7675eleq1d 2811 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → ((vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) ∈ ℝ ↔ (vol‘((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ ℝ))
7774, 76anbi12d 630 . . . . . . . . . . . . . . . . 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)))
7877cbvralvw 3225 . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 6893 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑙 → (𝑔𝑛) = (𝑔𝑙))
8281iundisj2 25566 . . . . . . . . . . . . . . 15 Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
83 disjeq2 5114 . . . . . . . . . . . . . . . 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 3057 . . . . . . . . . . . . . . 15 (Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ↔ Disj 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
8582, 84mpbir 230 . . . . . . . . . . . . . 14 Disj 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)
86 nnex 12264 . . . . . . . . . . . . . . . . 17 ℕ ∈ V
8786mptex 7232 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∈ V
88 fveq1 6892 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
8988eleq1d 2811 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((𝑓𝑛) ∈ dom vol ↔ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) ∈ dom vol))
9088fveq2d 6897 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → (vol‘(𝑓𝑛)) = (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)))
9190eleq1d 2811 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → ((vol‘(𝑓𝑛)) ∈ ℝ ↔ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) ∈ ℝ))
9289, 91anbi12d 630 . . . . . . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) = ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9594disjeq2dv 5115 . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . 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 5022 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙))) → 𝑛 ∈ ℕ (𝑓𝑛) = 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))
9897fveq2d 6897 . . . . . . . . . . . . . . . . . 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 14020 . . . . . . . . . . . . . . . . . . . . . . 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 2754 . . . . . . . . . . . . . . . . . . . . 21 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
104103rneqi 5935 . . . . . . . . . . . . . . . . . . . 20 ran 𝑆 = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛))))
105104supeq1i 9483 . . . . . . . . . . . . . . . . . . 19 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝑓𝑛)))), ℝ*, < )
10690mpteq2dv 5247 . . . . . . . . . . . . . . . . . . . . . 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 14023 . . . . . . . . . . . . . . . . . . . . 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 5936 . . . . . . . . . . . . . . . . . . . 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 9482 . . . . . . . . . . . . . . . . . . 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, 109eqtrid 2778 . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . 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 3537 . . . . . . . . . . . . . . 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 5014 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
116115fveq2i 6896 . . . . . . . . . . . . . . 15 (vol‘ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛)) = (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))
11766mpteq2ia 5248 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ (vol‘((𝑚 ∈ ℕ ↦ ((𝑔𝑚) ∖ 𝑙 ∈ (1..^𝑚)(𝑔𝑙)))‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))
118 seqeq3 14020 . . . . . . . . . . . . . . . . . 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 5935 . . . . . . . . . . . . . . . 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 9483 . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 25565 . . . . . . . . . . . . . . . 16 𝑛 ∈ ℕ (𝑔𝑛) = 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))
126 fofun 6808 . . . . . . . . . . . . . . . . 17 (𝑔:ℕ–onto𝐴 → Fun 𝑔)
127 funiunfv 7255 . . . . . . . . . . . . . . . . 17 (Fun 𝑔 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
128126, 127syl 17 . . . . . . . . . . . . . . . 16 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ (𝑔𝑛) = (𝑔 “ ℕ))
129125, 128eqtr3id 2780 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = (𝑔 “ ℕ))
13029unieqd 4918 . . . . . . . . . . . . . . 15 (𝑔:ℕ–onto𝐴 (𝑔 “ ℕ) = 𝐴)
131129, 130eqtrd 2766 . . . . . . . . . . . . . 14 (𝑔:ℕ–onto𝐴 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) = 𝐴)
132131fveq2d 6897 . . . . . . . . . . . . 13 (𝑔:ℕ–onto𝐴 → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
133132adantr 479 . . . . . . . . . . . 12 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → (vol‘ 𝑛 ∈ ℕ ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = (vol‘ 𝐴))
13456sseq1d 4010 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((𝑔𝑚) ⊆ ℝ ↔ (𝑔𝑛) ⊆ ℝ))
13556fveqeq2d 6901 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → ((vol*‘(𝑔𝑚)) = 0 ↔ (vol*‘(𝑔𝑛)) = 0))
136134, 135anbi12d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ↔ ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0)))
137136rspccva 3606 . . . . . . . . . . . . . . . . . . 19 ((∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) ∧ 𝑛 ∈ ℕ) → ((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0))
138 ssdifss 4132 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ⊆ ℝ → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
139138adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ)
140 difss 4128 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛)
141 ovolssnul 25504 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ (𝑔𝑛) ∧ (𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
142140, 141mp3an1 1445 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0)
143139, 142jca 510 . . . . . . . . . . . . . . . . . . . . 21 (((𝑔𝑛) ⊆ ℝ ∧ (vol*‘(𝑔𝑛)) = 0) → (((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0))
144 nulmbl 25552 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ⊆ ℝ ∧ (vol*‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))) = 0) → ((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)) ∈ dom vol)
145 mblvol 25547 . . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . 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 5245 . . . . . . . . . . . . . . . . 17 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙)))) = (𝑛 ∈ ℕ ↦ 0))
150149seqeq3d 14023 . . . . . . . . . . . . . . . 16 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = seq1( + , (𝑛 ∈ ℕ ↦ 0)))
151150rneqd 5936 . . . . . . . . . . . . . . 15 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))) = ran seq1( + , (𝑛 ∈ ℕ ↦ 0)))
152151supeq1d 9482 . . . . . . . . . . . . . 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 11247 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℂ
154 ser1const 14072 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ ℂ ∧ 𝑚 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
155153, 154mpan 688 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = (𝑚 · 0))
156 nncn 12266 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
157156mul01d 11454 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → (𝑚 · 0) = 0)
158155, 157eqtrd 2766 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑚) = 0)
159158mpteq2ia 5248 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)) = (𝑚 ∈ ℕ ↦ 0)
160 fconstmpt 5736 . . . . . . . . . . . . . . . . . . . . 21 (ℕ × {0}) = (𝑛 ∈ ℕ ↦ 0)
161 seqeq3 14020 . . . . . . . . . . . . . . . . . . . . 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 12638 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℤ
164 seqfn 14027 . . . . . . . . . . . . . . . . . . . . . 22 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
165163, 164ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
166 nnuz 12911 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
167166fneq2i 6650 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
168 dffn5 6953 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
169167, 168bitr3i 276 . . . . . . . . . . . . . . . . . . . . 21 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚)))
170165, 169mpbi 229 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (ℕ × {0})) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
171162, 170eqtr3i 2756 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (𝑚 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑚))
172 fconstmpt 5736 . . . . . . . . . . . . . . . . . . 19 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
173159, 171, 1723eqtr4i 2764 . . . . . . . . . . . . . . . . . 18 seq1( + , (𝑛 ∈ ℕ ↦ 0)) = (ℕ × {0})
174173rneqi 5935 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
175 1nn 12269 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
176 ne0i 4334 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℕ → ℕ ≠ ∅)
177 rnxp 6173 . . . . . . . . . . . . . . . . . 18 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
178175, 176, 177mp2b 10 . . . . . . . . . . . . . . . . 17 ran (ℕ × {0}) = {0}
179174, 178eqtri 2754 . . . . . . . . . . . . . . . 16 ran seq1( + , (𝑛 ∈ ℕ ↦ 0)) = {0}
180179supeq1i 9483 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
181 xrltso 13168 . . . . . . . . . . . . . . . 16 < Or ℝ*
182 0xr 11302 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
183 supsn 9508 . . . . . . . . . . . . . . . 16 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
184181, 182, 183mp2an 690 . . . . . . . . . . . . . . 15 sup({0}, ℝ*, < ) = 0
185180, 184eqtri 2754 . . . . . . . . . . . . . 14 sup(ran seq1( + , (𝑛 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
186152, 185eqtrdi 2782 . . . . . . . . . . . . 13 (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑔𝑛) ∖ 𝑙 ∈ (1..^𝑛)(𝑔𝑙))))), ℝ*, < ) = 0)
187186adantl 480 . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 ((𝑔:ℕ–onto𝐴 ∧ ∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0)) → 0 = (vol‘ 𝐴))
189188ex 411 . . . . . . . . . 10 (𝑔:ℕ–onto𝐴 → (∀𝑚 ∈ ℕ ((𝑔𝑚) ⊆ ℝ ∧ (vol*‘(𝑔𝑚)) = 0) → 0 = (vol‘ 𝐴)))
19038, 189sylbid 239 . . . . . . . . 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 1926 . . . . . . 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 452 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴)))
19511, 194pm2.61ine 3015 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol‘ 𝐴))
196 renepnf 11303 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
19747, 196mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
198 fveq2 6893 . . . . . . 7 ( 𝐴 = ℝ → (vol‘ 𝐴) = (vol‘ℝ))
199 rembl 25557 . . . . . . . . 9 ℝ ∈ dom vol
200 mblvol 25547 . . . . . . . . 9 (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ))
201199, 200ax-mp 5 . . . . . . . 8 (vol‘ℝ) = (vol*‘ℝ)
202 ovolre 25542 . . . . . . . 8 (vol*‘ℝ) = +∞
203201, 202eqtri 2754 . . . . . . 7 (vol‘ℝ) = +∞
204198, 203eqtrdi 2782 . . . . . 6 ( 𝐴 = ℝ → (vol‘ 𝐴) = +∞)
205197, 204neeqtrrd 3005 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol‘ 𝐴))
206205necon2i 2965 . . . 4 (0 = (vol‘ 𝐴) → 𝐴 ≠ ℝ)
207195, 206syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
208207expr 455 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
209 eqimss 4037 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
210209necon3bi 2957 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
211208, 210pm2.61d1 180 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  Vcvv 3462  cdif 3943  wss 3946  c0 4322  {csn 4623   cuni 4905   ciun 4993  Disj wdisj 5110   class class class wbr 5145  cmpt 5228   Or wor 5585   × cxp 5672  dom cdm 5674  ran crn 5675  cima 5677  Fun wfun 6540   Fn wfn 6541  ontowfo 6544  cfv 6546  (class class class)co 7416  cdom 8964  csdm 8965  supcsup 9476  cc 11147  cr 11148  0cc0 11149  1c1 11150   + caddc 11152   · cmul 11154  +∞cpnf 11286  *cxr 11288   < clt 11289  cn 12258  cz 12604  cuz 12868  ..^cfzo 13675  seqcseq 14015  vol*covol 25479  volcvol 25480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-inf2 9677  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-disj 5111  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fi 9447  df-sup 9478  df-inf 9479  df-oi 9546  df-dju 9937  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-n0 12519  df-z 12605  df-uz 12869  df-q 12979  df-rp 13023  df-xneg 13140  df-xadd 13141  df-xmul 13142  df-ioo 13376  df-ico 13378  df-icc 13379  df-fz 13533  df-fzo 13676  df-fl 13806  df-seq 14016  df-exp 14076  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-clim 15485  df-sum 15686  df-rest 17432  df-topgen 17453  df-psmet 21331  df-xmet 21332  df-met 21333  df-bl 21334  df-mopn 21335  df-top 22884  df-topon 22901  df-bases 22937  df-cmp 23379  df-ovol 25481  df-vol 25482
This theorem is referenced by: (None)
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