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Theorem itgitg1 23381
 Description: Transfer an integral using ∫1 to an equivalent integral using ∫. (Contributed by Mario Carneiro, 6-Aug-2014.)
Assertion
Ref Expression
itgitg1 (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹𝑥) d𝑥 = (∫1𝐹))
Distinct variable group:   𝑥,𝐹

Proof of Theorem itgitg1
StepHypRef Expression
1 i1ff 23249 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
21ffvelrnda 6267 . . 3 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
31feqmptd 6159 . . . 4 (𝐹 ∈ dom ∫1𝐹 = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
4 i1fibl 23380 . . . 4 (𝐹 ∈ dom ∫1𝐹 ∈ 𝐿1)
53, 4eqeltrrd 2689 . . 3 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹𝑥)) ∈ 𝐿1)
62, 5itgreval 23369 . 2 (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹𝑥) d𝑥 = (∫ℝif(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) d𝑥))
7 0re 9919 . . . . . . 7 0 ∈ ℝ
8 ifcl 4080 . . . . . . 7 (((𝐹𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ ℝ)
92, 7, 8sylancl 693 . . . . . 6 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ ℝ)
10 max1 11890 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
117, 2, 10sylancr 694 . . . . . 6 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
12 id 22 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹 ∈ dom ∫1)
133, 12eqeltrrd 2689 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹𝑥)) ∈ dom ∫1)
1413i1fposd 23280 . . . . . . 7 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ dom ∫1)
15 i1fibl 23380 . . . . . . 7 ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ 𝐿1)
1614, 15syl 17 . . . . . 6 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ 𝐿1)
179, 11, 16itgitg2 23379 . . . . 5 (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))))
1811ralrimiva 2949 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
19 reex 9906 . . . . . . . . . 10 ℝ ∈ V
2019a1i 11 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → ℝ ∈ V)
217a1i 11 . . . . . . . . 9 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → 0 ∈ ℝ)
22 fconstmpt 5085 . . . . . . . . . 10 (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
2322a1i 11 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0))
24 eqidd 2611 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
2520, 21, 9, 23, 24ofrfval2 6813 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ((ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ↔ ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
2618, 25mpbird 246 . . . . . . 7 (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
27 ax-resscn 9872 . . . . . . . . 9 ℝ ⊆ ℂ
2827a1i 11 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ℝ ⊆ ℂ)
29 eqid 2610 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
309, 29fmptd 6292 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)):ℝ⟶ℝ)
31 ffn 5958 . . . . . . . . 9 ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)):ℝ⟶ℝ → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) Fn ℝ)
3230, 31syl 17 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) Fn ℝ)
3328, 320pledm 23246 . . . . . . 7 (𝐹 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ↔ (ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))))
3426, 33mpbird 246 . . . . . 6 (𝐹 ∈ dom ∫1 → 0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
35 itg2itg1 23309 . . . . . 6 (((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ dom ∫1 ∧ 0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))))
3614, 34, 35syl2anc 691 . . . . 5 (𝐹 ∈ dom ∫1 → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))))
3717, 36eqtrd 2644 . . . 4 (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) d𝑥 = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))))
382renegcld 10336 . . . . . . 7 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → -(𝐹𝑥) ∈ ℝ)
39 ifcl 4080 . . . . . . 7 ((-(𝐹𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ ℝ)
4038, 7, 39sylancl 693 . . . . . 6 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ ℝ)
41 max1 11890 . . . . . . 7 ((0 ∈ ℝ ∧ -(𝐹𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
427, 38, 41sylancr 694 . . . . . 6 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
43 neg1rr 11002 . . . . . . . . . . . 12 -1 ∈ ℝ
4443a1i 11 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → -1 ∈ ℝ)
45 fconstmpt 5085 . . . . . . . . . . . 12 (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1)
4645a1i 11 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 → (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1))
4720, 44, 2, 46, 3offval2 6812 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘𝑓 · 𝐹) = (𝑥 ∈ ℝ ↦ (-1 · (𝐹𝑥))))
482recnd 9947 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℂ)
4948mulm1d 10361 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → (-1 · (𝐹𝑥)) = -(𝐹𝑥))
5049mpteq2dva 4672 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (-1 · (𝐹𝑥))) = (𝑥 ∈ ℝ ↦ -(𝐹𝑥)))
5147, 50eqtrd 2644 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘𝑓 · 𝐹) = (𝑥 ∈ ℝ ↦ -(𝐹𝑥)))
5243a1i 11 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 → -1 ∈ ℝ)
5312, 52i1fmulc 23276 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘𝑓 · 𝐹) ∈ dom ∫1)
5451, 53eqeltrrd 2689 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ -(𝐹𝑥)) ∈ dom ∫1)
5554i1fposd 23280 . . . . . . 7 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ dom ∫1)
56 i1fibl 23380 . . . . . . 7 ((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ 𝐿1)
5755, 56syl 17 . . . . . 6 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ 𝐿1)
5840, 42, 57itgitg2 23379 . . . . 5 (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
5942ralrimiva 2949 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
60 eqidd 2611 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
6120, 21, 40, 23, 60ofrfval2 6813 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ((ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ↔ ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
6259, 61mpbird 246 . . . . . . 7 (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
63 eqid 2610 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
6440, 63fmptd 6292 . . . . . . . . 9 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)):ℝ⟶ℝ)
65 ffn 5958 . . . . . . . . 9 ((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)):ℝ⟶ℝ → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) Fn ℝ)
6664, 65syl 17 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) Fn ℝ)
6728, 660pledm 23246 . . . . . . 7 (𝐹 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ↔ (ℝ × {0}) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
6862, 67mpbird 246 . . . . . 6 (𝐹 ∈ dom ∫1 → 0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
69 itg2itg1 23309 . . . . . 6 (((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ dom ∫1 ∧ 0𝑝𝑟 ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
7055, 68, 69syl2anc 691 . . . . 5 (𝐹 ∈ dom ∫1 → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
7158, 70eqtrd 2644 . . . 4 (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) d𝑥 = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
7237, 71oveq12d 6567 . . 3 (𝐹 ∈ dom ∫1 → (∫ℝif(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) d𝑥) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))))
73 itg1sub 23282 . . . 4 (((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∈ dom ∫1 ∧ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) ∈ dom ∫1) → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))))
7414, 55, 73syl2anc 691 . . 3 (𝐹 ∈ dom ∫1 → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))))
7572, 74eqtr4d 2647 . 2 (𝐹 ∈ dom ∫1 → (∫ℝif(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) d𝑥) = (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))))
76 max0sub 11901 . . . . . 6 ((𝐹𝑥) ∈ ℝ → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
772, 76syl 17 . . . . 5 ((𝐹 ∈ dom ∫1𝑥 ∈ ℝ) → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
7877mpteq2dva 4672 . . . 4 (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
7920, 9, 40, 24, 60offval2 6812 . . . 4 (𝐹 ∈ dom ∫1 → ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) = (𝑥 ∈ ℝ ↦ (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
8078, 79, 33eqtr4d 2654 . . 3 (𝐹 ∈ dom ∫1 → ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) = 𝐹)
8180fveq2d 6107 . 2 (𝐹 ∈ dom ∫1 → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)) ∘𝑓 − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))) = (∫1𝐹))
826, 75, 813eqtrd 2648 1 (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹𝑥) d𝑥 = (∫1𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ∘𝑟 cofr 6794  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   ≤ cle 9954   − cmin 10145  -cneg 10146  ∫1citg1 23190  ∫2citg2 23191  𝐿1cibl 23192  ∫citg 23193  0𝑝c0p 23242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-itg 23198  df-0p 23243 This theorem is referenced by: (None)
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