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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  –onto→wfo 5924  ‘cfv 5926  (class class class)co 6690   ≼ cdom 7995   ≺ csdm 7996  supcsup 8387  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112  ℕcn 11058  ℤcz 11415  ℤ≥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)
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