MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fd Structured version   Visualization version   GIF version

Theorem i1fd 25609
Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fd.1 (𝜑𝐹:ℝ⟶ℝ)
i1fd.2 (𝜑 → ran 𝐹 ∈ Fin)
i1fd.3 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)
i1fd.4 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
Assertion
Ref Expression
i1fd (𝜑𝐹 ∈ dom ∫1)
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥

Proof of Theorem i1fd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 i1fd.1 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
21ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3 ffun 6725 . . . . . . . 8 (𝐹:ℝ⟶ℝ → Fun 𝐹)
4 funcnvcnv 6620 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
5 imadif 6637 . . . . . . . 8 (Fun 𝐹 → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))))
62, 3, 4, 54syl 19 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))))
7 ioof 13456 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8 frn 6729 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
97, 8ax-mp 5 . . . . . . . . . . . 12 ran (,) ⊆ 𝒫 ℝ
109sseli 3976 . . . . . . . . . . 11 (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
1110elpwid 4612 . . . . . . . . . 10 (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
1211ad2antlr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13 dfss4 4259 . . . . . . . . 9 (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1412, 13sylib 217 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1514imaeq2d 6063 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (𝐹𝑥))
166, 15eqtr3d 2770 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) = (𝐹𝑥))
17 fimacnv 6745 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → (𝐹 “ ℝ) = ℝ)
182, 17syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) = ℝ)
19 rembl 25468 . . . . . . . 8 ℝ ∈ dom vol
2018, 19eqeltrdi 2837 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) ∈ dom vol)
211adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22 inpreima 7073 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 “ (𝑦 ∩ ran 𝐹)) = ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)))
23 iunid 5063 . . . . . . . . . . . . . . . 16 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
2423imaeq2i 6061 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (𝐹 “ (𝑦 ∩ ran 𝐹))
25 imaiun 7255 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
2624, 25eqtr3i 2758 . . . . . . . . . . . . . 14 (𝐹 “ (𝑦 ∩ ran 𝐹)) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
27 cnvimass 6085 . . . . . . . . . . . . . . . 16 (𝐹𝑦) ⊆ dom 𝐹
28 cnvimarndm 6086 . . . . . . . . . . . . . . . 16 (𝐹 “ ran 𝐹) = dom 𝐹
2927, 28sseqtrri 4017 . . . . . . . . . . . . . . 15 (𝐹𝑦) ⊆ (𝐹 “ ran 𝐹)
30 df-ss 3964 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦))
3129, 30mpbi 229 . . . . . . . . . . . . . 14 ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦)
3222, 26, 313eqtr3g 2791 . . . . . . . . . . . . 13 (Fun 𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) = (𝐹𝑦))
3321, 3, 323syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) = (𝐹𝑦))
34 i1fd.2 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ∈ Fin)
3534adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36 inss2 4230 . . . . . . . . . . . . . 14 (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37 ssfi 9197 . . . . . . . . . . . . . 14 ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
3835, 36, 37sylancl 585 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39 simpll 766 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40 elinel1 4195 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
4140con3i 154 . . . . . . . . . . . . . . . . . . 19 (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4241adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43 disjsn 4716 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4442, 43sylibr 233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45 reldisj 4452 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∩ ran 𝐹) ⊆ ran 𝐹 → (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0})))
4636, 45ax-mp 5 . . . . . . . . . . . . . . . . 17 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4744, 46sylib 217 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4847sselda 3980 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝑥 ∈ (ran 𝐹 ∖ {0}))
49 i1fd.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)
5039, 48, 49syl2anc 583 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ∈ dom vol)
5150ralrimiva 3143 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
52 finiunmbl 25472 . . . . . . . . . . . . 13 (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5338, 51, 52syl2anc 583 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5433, 53eqeltrrd 2830 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ∈ dom vol)
5554ex 412 . . . . . . . . . 10 (𝜑 → (¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5655alrimiv 1923 . . . . . . . . 9 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5756ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
58 elndif 4127 . . . . . . . . 9 (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
5958adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60 reex 11229 . . . . . . . . . 10 ℝ ∈ V
6160difexi 5330 . . . . . . . . 9 (ℝ ∖ 𝑥) ∈ V
62 eleq2 2818 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
6362notbid 318 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64 imaeq2 6059 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ 𝑥)))
6564eleq1d 2814 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → ((𝐹𝑦) ∈ dom vol ↔ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
6663, 65imbi12d 344 . . . . . . . . 9 (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
6761, 66spcv 3592 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
6857, 59, 67sylc 65 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
69 difmbl 25471 . . . . . . 7 (((𝐹 “ ℝ) ∈ dom vol ∧ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7020, 68, 69syl2anc 583 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7116, 70eqeltrrd 2830 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
72 eleq2 2818 . . . . . . . . . . 11 (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
7372notbid 318 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74 imaeq2 6059 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7574eleq1d 2814 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ dom vol ↔ (𝐹𝑥) ∈ dom vol))
7673, 75imbi12d 344 . . . . . . . . 9 (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol)))
7776spvv 1993 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7856, 77syl 17 . . . . . . 7 (𝜑 → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7978imp 406 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8079adantlr 714 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8171, 80pm2.61dan 812 . . . 4 ((𝜑𝑥 ∈ ran (,)) → (𝐹𝑥) ∈ dom vol)
8281ralrimiva 3143 . . 3 (𝜑 → ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol)
83 ismbf 25556 . . . 4 (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
841, 83syl 17 . . 3 (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
8582, 84mpbird 257 . 2 (𝜑𝐹 ∈ MblFn)
86 mblvol 25458 . . . . . . . 8 ((𝐹𝑦) ∈ dom vol → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
8754, 86syl 17 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
88 mblss 25459 . . . . . . . . 9 ((𝐹𝑦) ∈ dom vol → (𝐹𝑦) ⊆ ℝ)
8954, 88syl 17 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ⊆ ℝ)
90 mblvol 25458 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) ∈ dom vol → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
9150, 90syl 17 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
92 i1fd.4 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9339, 48, 92syl2anc 583 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9491, 93eqeltrrd 2830 . . . . . . . . 9 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9538, 94fsumrecl 15712 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9633fveq2d 6901 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) = (vol*‘(𝐹𝑦)))
97 mblss 25459 . . . . . . . . . . . . 13 ((𝐹 “ {𝑥}) ∈ dom vol → (𝐹 “ {𝑥}) ⊆ ℝ)
9850, 97syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ⊆ ℝ)
9998, 94jca 511 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
10099ralrimiva 3143 . . . . . . . . . 10 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
101 ovolfiniun 25429 . . . . . . . . . 10 (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
10238, 100, 101syl2anc 583 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
10396, 102eqbrtrrd 5172 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
104 ovollecl 25411 . . . . . . . 8 (((𝐹𝑦) ⊆ ℝ ∧ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ ∧ (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥}))) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10589, 95, 103, 104syl3anc 1369 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10687, 105eqeltrd 2829 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) ∈ ℝ)
107106ex 412 . . . . 5 (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
108107alrimiv 1923 . . . 4 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
109 neldifsn 4796 . . . 4 ¬ 0 ∈ (ℝ ∖ {0})
11060difexi 5330 . . . . 5 (ℝ ∖ {0}) ∈ V
111 eleq2 2818 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112111notbid 318 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113 imaeq2 6059 . . . . . . . 8 (𝑦 = (ℝ ∖ {0}) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ {0})))
114113fveq2d 6901 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (vol‘(𝐹𝑦)) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
115114eleq1d 2814 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → ((vol‘(𝐹𝑦)) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
116112, 115imbi12d 344 . . . . 5 (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
117110, 116spcv 3592 . . . 4 (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
118108, 109, 117mpisyl 21 . . 3 (𝜑 → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
1191, 34, 1183jca 1126 . 2 (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
120 isi1f 25602 . 2 (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
12185, 119, 120sylanbrc 582 1 (𝜑𝐹 ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  wral 3058  cdif 3944  cin 3946  wss 3947  c0 4323  𝒫 cpw 4603  {csn 4629   ciun 4996   class class class wbr 5148   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  Fun wfun 6542  wf 6544  cfv 6548  Fincfn 8963  cr 11137  0cc0 11138  *cxr 11277  cle 11279  (,)cioo 13356  Σcsu 15664  vol*covol 25390  volcvol 25391  MblFncmbf 25542  1citg1 25543
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-map 8846  df-pm 8847  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9465  df-inf 9466  df-oi 9533  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-q 12963  df-rp 13007  df-xadd 13125  df-ioo 13360  df-ico 13362  df-icc 13363  df-fz 13517  df-fzo 13660  df-fl 13789  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-clim 15464  df-sum 15665  df-xmet 21271  df-met 21272  df-ovol 25392  df-vol 25393  df-mbf 25547  df-itg1 25548
This theorem is referenced by:  i1f0  25615  i1f1  25618  i1fadd  25623  i1fmul  25624  i1fmulc  25632  i1fres  25634  mbfi1fseqlem4  25647  itg2addnclem2  37145  ftc1anclem3  37168
  Copyright terms: Public domain W3C validator