Proof of Theorem caragenuncllem
Step | Hyp | Ref
| Expression |
1 | | caragenuncllem.o |
. . . . . 6
⊢ (𝜑 → 𝑂 ∈ OutMeas) |
2 | | caragenuncllem.s |
. . . . . 6
⊢ 𝑆 = (CaraGen‘𝑂) |
3 | | caragenuncllem.x |
. . . . . 6
⊢ 𝑋 = ∪
dom 𝑂 |
4 | | caragenuncllem.e |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ 𝑆) |
5 | | caragenuncllem.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
6 | 5 | ssinss1d 42577 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋) |
7 | 1, 2, 3, 4, 6 | caragensplit 44019 |
. . . . 5
⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹)))) |
8 | 7 | eqcomd 2744 |
. . . 4
⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)))) |
9 | | inass 4153 |
. . . . . . . 8
⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) |
10 | | incom 4134 |
. . . . . . . . . 10
⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹)) |
11 | | inabs 4189 |
. . . . . . . . . 10
⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸 |
12 | 10, 11 | eqtri 2766 |
. . . . . . . . 9
⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸 |
13 | 12 | ineq2i 4143 |
. . . . . . . 8
⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸) |
14 | 9, 13 | eqtri 2766 |
. . . . . . 7
⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸) |
15 | 14 | fveq2i 6769 |
. . . . . 6
⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸)) |
16 | | incom 4134 |
. . . . . . . . . 10
⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸)) |
17 | | indifcom 4206 |
. . . . . . . . . 10
⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸)) |
18 | 16, 17 | eqtr2i 2767 |
. . . . . . . . 9
⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹) |
19 | 18 | eqcomi 2747 |
. . . . . . . 8
⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸)) |
20 | | difundir 4214 |
. . . . . . . . . 10
⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) |
21 | | difid 4304 |
. . . . . . . . . . 11
⊢ (𝐸 ∖ 𝐸) = ∅ |
22 | 21 | uneq1i 4092 |
. . . . . . . . . 10
⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸)) |
23 | | 0un 4326 |
. . . . . . . . . 10
⊢ (∅
∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸) |
24 | 20, 22, 23 | 3eqtrri 2771 |
. . . . . . . . 9
⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸) |
25 | 24 | ineq2i 4143 |
. . . . . . . 8
⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) |
26 | | indif2 4204 |
. . . . . . . 8
⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) |
27 | 19, 25, 26 | 3eqtrri 2771 |
. . . . . . 7
⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹) |
28 | 27 | fveq2i 6769 |
. . . . . 6
⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) |
29 | 15, 28 | oveq12i 7279 |
. . . . 5
⊢ ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) |
30 | 29 | a1i 11 |
. . . 4
⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)))) |
31 | | eqidd 2739 |
. . . 4
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)))) |
32 | 8, 30, 31 | 3eqtrd 2782 |
. . 3
⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)))) |
33 | | difun1 4223 |
. . . . 5
⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹) |
34 | 33 | fveq2i 6769 |
. . . 4
⊢ (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹))) = (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) |
35 | 34 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹))) = (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) |
36 | 32, 35 | oveq12d 7285 |
. 2
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) |
37 | 5 | ssinss1d 42577 |
. . . . 5
⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋) |
38 | 1, 3, 37 | omexrcl 44026 |
. . . 4
⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈
ℝ*) |
39 | 1, 3, 37 | omecl 44022 |
. . . . 5
⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞)) |
40 | 39 | xrge0nemnfd 42852 |
. . . 4
⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞) |
41 | 38, 40 | jca 512 |
. . 3
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞)) |
42 | | caragenuncllem.f |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ 𝑆) |
43 | 1, 2, 42, 3 | caragenelss 44020 |
. . . . . 6
⊢ (𝜑 → 𝐹 ⊆ 𝑋) |
44 | 43 | ssinss2d 42589 |
. . . . 5
⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋) |
45 | 1, 3, 44 | omexrcl 44026 |
. . . 4
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈
ℝ*) |
46 | 1, 3, 44 | omecl 44022 |
. . . . 5
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞)) |
47 | 46 | xrge0nemnfd 42852 |
. . . 4
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞) |
48 | 45, 47 | jca 512 |
. . 3
⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞)) |
49 | 5 | ssdifssd 4076 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋) |
50 | 49 | ssdifssd 4076 |
. . . . 5
⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋) |
51 | 1, 3, 50 | omexrcl 44026 |
. . . 4
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈
ℝ*) |
52 | 1, 3, 50 | omecl 44022 |
. . . . 5
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞)) |
53 | 52 | xrge0nemnfd 42852 |
. . . 4
⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞) |
54 | 51, 53 | jca 512 |
. . 3
⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)) |
55 | | xaddass 12993 |
. . 3
⊢ ((((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞) ∧ ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞) ∧ ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)) → (((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))))) |
56 | 41, 48, 54, 55 | syl3anc 1370 |
. 2
⊢ (𝜑 → (((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))))) |
57 | 1, 2, 3, 42, 49 | caragensplit 44019 |
. . . 4
⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸))) |
58 | 57 | oveq2d 7283 |
. . 3
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
59 | 1, 2, 3, 4, 5 | caragensplit 44019 |
. . 3
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) |
60 | 58, 59 | eqtrd 2778 |
. 2
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴)) |
61 | 36, 56, 60 | 3eqtrd 2782 |
1
⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴)) |