Step | Hyp | Ref
| Expression |
1 | | meaiininclem.k |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) |
2 | | uzss 11708 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈
(ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) |
3 | 1, 2 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
(ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) |
4 | | meaiininclem.z |
. . . . . . . . . . . . 13
⊢ 𝑍 =
(ℤ≥‘𝑁) |
5 | 3, 4 | syl6sseqr 3652 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(ℤ≥‘𝐾) ⊆ 𝑍) |
6 | 5 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) →
(ℤ≥‘𝐾) ⊆ 𝑍) |
7 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → 𝑛 ∈ (ℤ≥‘𝐾)) |
8 | 6, 7 | sseldd 3604 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → 𝑛 ∈ 𝑍) |
9 | | meaiininclem.g |
. . . . . . . . . . . 12
⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
10 | 9 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))) |
11 | | meaiininclem.m |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ Meas) |
12 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ dom 𝑀 = dom 𝑀 |
13 | 11, 12 | dmmeasal 40669 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
14 | 13 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg) |
15 | 1, 4 | syl6eleqr 2712 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ 𝑍) |
16 | | meaiininclem.e |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
17 | 16 | ffvelrnda 6359 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀) |
18 | 15, 17 | mpdan 702 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀) |
19 | 18 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀) |
20 | 16 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀) |
21 | | saldifcl2 40546 |
. . . . . . . . . . . . 13
⊢ ((dom
𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀) |
22 | 14, 19, 20, 21 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀) |
23 | 22 | elexd 3214 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V) |
24 | 10, 23 | fvmpt2d 6293 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
25 | 8, 24 | syldan 487 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
26 | 25 | fveq2d 6195 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛)))) |
27 | 11 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → 𝑀 ∈ Meas) |
28 | 18 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀) |
29 | | meaiininclem.r |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) |
30 | 29 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) |
31 | 8, 20 | syldan 487 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝑛) ∈ dom 𝑀) |
32 | | simpl 473 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑) |
33 | 32, 5 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ≥‘𝐾) ⊆ 𝑍) |
34 | | elfzouz 12474 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ≥‘𝐾)) |
35 | 34 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ≥‘𝐾)) |
36 | 33, 35 | sseldd 3604 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍) |
37 | | eleq1 2689 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍)) |
38 | 37 | anbi2d 740 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍))) |
39 | | oveq1 6657 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1)) |
40 | 39 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1))) |
41 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚)) |
42 | 40, 41 | sseq12d 3634 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))) |
43 | 38, 42 | imbi12d 334 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) ↔ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))) |
44 | | meaiininclem.i |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) |
45 | 43, 44 | chvarv 2263 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
46 | 32, 36, 45 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
47 | 46 | adantlr 751 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
48 | 7, 47 | ssdec 39265 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
49 | 27, 28, 30, 31, 48 | meadif 40696 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) |
50 | 26, 49 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) |
51 | 50 | oveq2d 6666 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))) |
52 | 29 | recnd 10068 |
. . . . . . . 8
⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ) |
53 | 52 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ) |
54 | 27, 28, 30, 48, 31 | meassre 40694 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ) |
55 | 54 | recnd 10068 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ) |
56 | 53, 55 | nncand 10397 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛))) |
57 | 51, 56 | eqtr2d 2657 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) |
58 | 57 | mpteq2dva 4744 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))) |
59 | | nfv 1843 |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
60 | | eqid 2622 |
. . . . 5
⊢
(ℤ≥‘𝐾) = (ℤ≥‘𝐾) |
61 | 1 | eluzelzd 39591 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ℤ) |
62 | | difssd 3738 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾)) |
63 | 24, 62 | eqsstrd 3639 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
64 | 8, 63 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
65 | 22, 9 | fmptd 6385 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀) |
66 | 65 | ffvelrnda 6359 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ dom 𝑀) |
67 | 8, 66 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) ∈ dom 𝑀) |
68 | 27, 28, 30, 64, 67 | meassre 40694 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ) |
69 | 68 | recnd 10068 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ) |
70 | | meaiininclem.n |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
71 | 44 | sscond 3747 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
72 | 41 | difeq2d 3728 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
73 | 72 | cbvmptv 4750 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
74 | 9, 73 | eqtri 2644 |
. . . . . . . . . . . 12
⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
75 | 74 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))) |
76 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1))) |
77 | 76 | difeq2d 3728 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
78 | 77 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 = (𝑛 + 1)) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
79 | 4 | peano2uzs 11742 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍) |
80 | 79 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍) |
81 | | fvex 6201 |
. . . . . . . . . . . . 13
⊢ (𝐸‘𝐾) ∈ V |
82 | 81 | difexi 4809 |
. . . . . . . . . . . 12
⊢ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V |
83 | 82 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V) |
84 | 75, 78, 80, 83 | fvmptd 6288 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
85 | 24, 84 | sseq12d 3634 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)) ↔ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))) |
86 | 71, 85 | mpbird 247 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1))) |
87 | 11 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas) |
88 | 87, 12, 66, 19, 63 | meassle 40680 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐺‘𝑛)) ≤ (𝑀‘(𝐸‘𝐾))) |
89 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) |
90 | 11, 70, 4, 65, 86, 29, 88, 89 | meaiuninc2 40699 |
. . . . . . 7
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛))) |
91 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) |
92 | 4, 89, 15, 91 | climresmpt 39891 |
. . . . . . 7
⊢ (𝜑 → ((𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)))) |
93 | 90, 92 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛))) |
94 | | meaiininclem.f |
. . . . . . . . 9
⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) |
95 | 94 | eqcomi 2631 |
. . . . . . . 8
⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹 |
96 | 95 | fveq2i 6194 |
. . . . . . 7
⊢ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹) |
97 | 96 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹)) |
98 | 93, 97 | breqtrd 4679 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹)) |
99 | 59, 60, 61, 52, 69, 98 | climsubc1mpt 39894 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
100 | 58, 99 | eqbrtrd 4675 |
. . 3
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
101 | | eqid 2622 |
. . . 4
⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
102 | | eqid 2622 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) |
103 | 4, 101, 15, 102 | climresmpt 39891 |
. . 3
⊢ (𝜑 → ((𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))) |
104 | 100, 103 | mpbid 222 |
. 2
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
105 | | meaiininclem.s |
. . . 4
⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
106 | 105 | a1i 11 |
. . 3
⊢ (𝜑 → 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))) |
107 | | eqidd 2623 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)))) |
108 | 4 | uzct 39232 |
. . . . . . . . . 10
⊢ 𝑍 ≼
ω |
109 | 108 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ≼ ω) |
110 | 13, 109, 66 | saliuncl 40542 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀) |
111 | 94, 110 | syl5eqel 2705 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ dom 𝑀) |
112 | | saldifcl2 40546 |
. . . . . . . 8
⊢ ((dom
𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀) |
113 | 13, 18, 111, 112 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀) |
114 | | disjdif 4040 |
. . . . . . . 8
⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅ |
115 | 114 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅) |
116 | 63 | iunssd 39271 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
117 | 94, 116 | syl5eqss 3649 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾)) |
118 | 11, 18, 29, 117, 111 | meassre 40694 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ) |
119 | | difssd 3738 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾)) |
120 | 11, 18, 29, 119, 113 | meassre 40694 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ) |
121 | 11, 12, 111, 113, 115, 118, 120 | meadjunre 40693 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹)))) |
122 | | undif 4049 |
. . . . . . . 8
⊢ (𝐹 ⊆ (𝐸‘𝐾) ↔ (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾)) |
123 | 117, 122 | sylib 208 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾)) |
124 | 123 | fveq2d 6195 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))) |
125 | 107, 121,
124 | 3eqtr3d 2664 |
. . . . 5
⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))) |
126 | 118 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ) |
127 | 120 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ) |
128 | 52, 126, 127 | subaddd 10410 |
. . . . 5
⊢ (𝜑 → (((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ↔ ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))) |
129 | 125, 128 | mpbird 247 |
. . . 4
⊢ (𝜑 → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) |
130 | | simpllr 799 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
131 | | simplr 792 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑛 ∈ 𝑍) |
132 | | eldifi 3732 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾)) |
133 | 132 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾)) |
134 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ (𝐸‘𝑛)) |
135 | 133, 134 | eldifd 3585 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
136 | | rspe 3003 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
137 | 131, 135,
136 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
138 | | eliun 4524 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
139 | 137, 138 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
140 | 139 | adantlll 754 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
141 | 94 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) |
142 | 24 | iuneq2dv 4542 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
143 | 141, 142 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
144 | 143 | eqcomd 2628 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹) |
145 | 144 | ad3antrrr 766 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹) |
146 | 140, 145 | eleqtrd 2703 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹) |
147 | | elndif 3734 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
148 | 146, 147 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
149 | 130, 148 | condan 835 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛)) |
150 | 149 | ralrimiva 2966 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛)) |
151 | | vex 3203 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
152 | | eliin 4525 |
. . . . . . . . 9
⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))) |
153 | 151, 152 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛)) |
154 | 150, 153 | sylibr 224 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
155 | 154 | ssd 39252 |
. . . . . 6
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
156 | | ssid 3624 |
. . . . . . . . . . . . 13
⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾) |
157 | 156 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) |
158 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾)) |
159 | 158 | sseq1d 3632 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾))) |
160 | 159 | rspcev 3309 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ 𝑍 ∧ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
161 | 15, 157, 160 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
162 | | iinss 4571 |
. . . . . . . . . . 11
⊢
(∃𝑛 ∈
𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
163 | 161, 162 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
164 | 163 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
165 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
166 | 164, 165 | sseldd 3604 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾)) |
167 | | nfcv 2764 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑥 |
168 | | nfii1 4551 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
169 | 167, 168 | nfel 2777 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
170 | | iinss2 4572 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑍 → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛)) |
171 | 170 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛)) |
172 | | simpl 473 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
173 | 171, 172 | sseldd 3604 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛)) |
174 | | elndif 3734 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝐸‘𝑛) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
175 | 173, 174 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
176 | 175 | ex 450 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))) |
177 | 169, 176 | ralrimi 2957 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
178 | | ralnex 2992 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
179 | 177, 178 | sylib 208 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
180 | 179, 138 | sylnibr 319 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
181 | 180 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
182 | 143 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
183 | 181, 182 | neleqtrrd 2723 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ 𝐹) |
184 | 166, 183 | eldifd 3585 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
185 | 184 | ssd 39252 |
. . . . . 6
⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ((𝐸‘𝐾) ∖ 𝐹)) |
186 | 155, 185 | eqssd 3620 |
. . . . 5
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
187 | 186 | fveq2d 6195 |
. . . 4
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
188 | 129, 187 | eqtr2d 2657 |
. . 3
⊢ (𝜑 → (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
189 | 106, 188 | breq12d 4666 |
. 2
⊢ (𝜑 → (𝑆 ⇝ (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))) |
190 | 104, 189 | mpbird 247 |
1
⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛))) |