| Step | Hyp | Ref
| Expression |
| 1 | | meaiininclem.k |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) |
| 2 | | uzss 12901 |
. . . . . . . . . . . . . 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 | sseqtrrdi 4025 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(ℤ≥‘𝐾) ⊆ 𝑍) |
| 6 | 5 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) →
(ℤ≥‘𝐾) ⊆ 𝑍) |
| 7 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → 𝑛 ∈ (ℤ≥‘𝐾)) |
| 8 | 6, 7 | sseldd 3984 |
. . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . 15
⊢ dom 𝑀 = dom 𝑀 |
| 13 | 11, 12 | dmmeasal 46467 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
| 14 | 13 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg) |
| 15 | 1, 4 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ 𝑍) |
| 16 | | meaiininclem.e |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
| 17 | 16 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀) |
| 18 | 15, 17 | mpdan 687 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀) |
| 19 | 18 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀) |
| 20 | 16 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀) |
| 21 | | saldifcl2 46343 |
. . . . . . . . . . . . 13
⊢ ((dom
𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀) |
| 22 | 14, 19, 20, 21 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀) |
| 23 | 22 | elexd 3504 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V) |
| 24 | 10, 23 | fvmpt2d 7029 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 25 | 8, 24 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 26 | 25 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛)))) |
| 27 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → 𝑀 ∈ Meas) |
| 28 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀) |
| 29 | | meaiininclem.r |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) |
| 30 | 29 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) |
| 31 | 8, 20 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝑛) ∈ dom 𝑀) |
| 32 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑) |
| 33 | 32, 5 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ≥‘𝐾) ⊆ 𝑍) |
| 34 | | elfzouz 13703 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ≥‘𝐾)) |
| 35 | 34 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ≥‘𝐾)) |
| 36 | 33, 35 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍) |
| 37 | | eleq1w 2824 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍)) |
| 38 | 37 | anbi2d 630 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍))) |
| 39 | | fvoveq1 7454 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1))) |
| 40 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚)) |
| 41 | 39, 40 | sseq12d 4017 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))) |
| 42 | 38, 41 | imbi12d 344 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) ↔ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))) |
| 43 | | meaiininclem.i |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) |
| 44 | 42, 43 | chvarvv 1998 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
| 45 | 32, 36, 44 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
| 46 | 45 | adantlr 715 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)) |
| 47 | 7, 46 | ssdec 45093 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 48 | 27, 28, 30, 31, 47 | meadif 46494 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) |
| 49 | 26, 48 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) |
| 50 | 49 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))) |
| 51 | 29 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ) |
| 52 | 51 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ) |
| 53 | 27, 28, 30, 47, 31 | meassre 46492 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ) |
| 54 | 53 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ) |
| 55 | 52, 54 | nncand 11625 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛))) |
| 56 | 50, 55 | eqtr2d 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) |
| 57 | 56 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))) |
| 58 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
| 59 | | eqid 2737 |
. . . . 5
⊢
(ℤ≥‘𝐾) = (ℤ≥‘𝐾) |
| 60 | 1 | eluzelzd 45386 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ℤ) |
| 61 | | difssd 4137 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾)) |
| 62 | 24, 61 | eqsstrd 4018 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
| 63 | 8, 62 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
| 64 | 22, 9 | fmptd 7134 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀) |
| 65 | 64 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ dom 𝑀) |
| 66 | 8, 65 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑛) ∈ dom 𝑀) |
| 67 | 27, 28, 30, 63, 66 | meassre 46492 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ) |
| 68 | 67 | recnd 11289 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ) |
| 69 | | meaiininclem.n |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 70 | 43 | sscond 4146 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
| 71 | 40 | difeq2d 4126 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
| 72 | 71 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
| 73 | 9, 72 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚))) |
| 74 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1))) |
| 75 | 74 | difeq2d 4126 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
| 76 | 4 | peano2uzs 12944 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍) |
| 77 | 76 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍) |
| 78 | | fvex 6919 |
. . . . . . . . . . . . 13
⊢ (𝐸‘𝐾) ∈ V |
| 79 | 78 | difexi 5330 |
. . . . . . . . . . . 12
⊢ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V |
| 80 | 79 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V) |
| 81 | 73, 75, 77, 80 | fvmptd3 7039 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))) |
| 82 | 24, 81 | sseq12d 4017 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)) ↔ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))) |
| 83 | 70, 82 | mpbird 257 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1))) |
| 84 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas) |
| 85 | 84, 12, 65, 19, 62 | meassle 46478 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐺‘𝑛)) ≤ (𝑀‘(𝐸‘𝐾))) |
| 86 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) |
| 87 | 11, 69, 4, 64, 83, 29, 85, 86 | meaiuninc2 46497 |
. . . . . . 7
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛))) |
| 88 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) |
| 89 | 4, 86, 15, 88 | climresmpt 45674 |
. . . . . . 7
⊢ (𝜑 → ((𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)))) |
| 90 | 87, 89 | mpbird 257 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛))) |
| 91 | | meaiininclem.f |
. . . . . . . . 9
⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) |
| 92 | 91 | eqcomi 2746 |
. . . . . . . 8
⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹 |
| 93 | 92 | fveq2i 6909 |
. . . . . . 7
⊢ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹) |
| 94 | 93 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑀‘∪
𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹)) |
| 95 | 90, 94 | breqtrd 5169 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹)) |
| 96 | 58, 59, 60, 51, 68, 95 | climsubc1mpt 45677 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
| 97 | 57, 96 | eqbrtrd 5165 |
. . 3
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
| 98 | | eqid 2737 |
. . . 4
⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
| 99 | | eqid 2737 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) |
| 100 | 4, 98, 15, 99 | climresmpt 45674 |
. . 3
⊢ (𝜑 → ((𝑛 ∈ (ℤ≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))) |
| 101 | 97, 100 | mpbid 232 |
. 2
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
| 102 | | meaiininclem.s |
. . . 4
⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
| 103 | 102 | a1i 11 |
. . 3
⊢ (𝜑 → 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))) |
| 104 | | eqidd 2738 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)))) |
| 105 | 4 | uzct 45068 |
. . . . . . . . . 10
⊢ 𝑍 ≼
ω |
| 106 | 105 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ≼ ω) |
| 107 | 13, 106, 65 | saliuncl 46338 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀) |
| 108 | 91, 107 | eqeltrid 2845 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ dom 𝑀) |
| 109 | | saldifcl2 46343 |
. . . . . . . 8
⊢ ((dom
𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀) |
| 110 | 13, 18, 108, 109 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀) |
| 111 | | disjdif 4472 |
. . . . . . . 8
⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅ |
| 112 | 111 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅) |
| 113 | 62 | iunssd 5050 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾)) |
| 114 | 91, 113 | eqsstrid 4022 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾)) |
| 115 | 11, 18, 29, 114, 108 | meassre 46492 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ) |
| 116 | | difssd 4137 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾)) |
| 117 | 11, 18, 29, 116, 110 | meassre 46492 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ) |
| 118 | 11, 12, 108, 110, 112, 115, 117 | meadjunre 46491 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹)))) |
| 119 | | undif 4482 |
. . . . . . . 8
⊢ (𝐹 ⊆ (𝐸‘𝐾) ↔ (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾)) |
| 120 | 114, 119 | sylib 218 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾)) |
| 121 | 120 | fveq2d 6910 |
. . . . . 6
⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))) |
| 122 | 104, 118,
121 | 3eqtr3d 2785 |
. . . . 5
⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))) |
| 123 | 115 | recnd 11289 |
. . . . . 6
⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ) |
| 124 | 117 | recnd 11289 |
. . . . . 6
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ) |
| 125 | 51, 123, 124 | subaddd 11638 |
. . . . 5
⊢ (𝜑 → (((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ↔ ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))) |
| 126 | 122, 125 | mpbird 257 |
. . . 4
⊢ (𝜑 → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) |
| 127 | | simpllr 776 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
| 128 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑛 ∈ 𝑍) |
| 129 | | eldifi 4131 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾)) |
| 130 | 129 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾)) |
| 131 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ (𝐸‘𝑛)) |
| 132 | 130, 131 | eldifd 3962 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 133 | | rspe 3249 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 134 | 128, 132,
133 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 135 | | eliun 4995 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 136 | 134, 135 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 137 | 136 | adantlll 718 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 138 | 91 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) |
| 139 | 24 | iuneq2dv 5016 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 140 | 138, 139 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 141 | 140 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹) |
| 142 | 141 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹) |
| 143 | 137, 142 | eleqtrd 2843 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹) |
| 144 | | elndif 4133 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
| 145 | 143, 144 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
| 146 | 127, 145 | condan 818 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛)) |
| 147 | 146 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛)) |
| 148 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
| 149 | | eliin 4996 |
. . . . . . . . 9
⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))) |
| 150 | 148, 149 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛)) |
| 151 | 147, 150 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 152 | 151 | ssd 45085 |
. . . . . 6
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 153 | | ssid 4006 |
. . . . . . . . . . . 12
⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾) |
| 154 | 153 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) |
| 155 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾)) |
| 156 | 155 | sseq1d 4015 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾))) |
| 157 | 156 | rspcev 3622 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ 𝑍 ∧ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 158 | 15, 154, 157 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 159 | | iinss 5056 |
. . . . . . . . . 10
⊢
(∃𝑛 ∈
𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 160 | 158, 159 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 161 | 160 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾)) |
| 162 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 163 | 161, 162 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾)) |
| 164 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛𝑥 |
| 165 | | nfii1 5029 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
| 166 | 164, 165 | nfel 2920 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
| 167 | | iinss2 5057 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛)) |
| 168 | 167 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛)) |
| 169 | | simpl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 170 | 168, 169 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛)) |
| 171 | | elndif 4133 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (𝐸‘𝑛) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 172 | 170, 171 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 173 | 172 | ex 412 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))) |
| 174 | 166, 173 | ralrimi 3257 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 175 | | ralnex 3072 |
. . . . . . . . . . 11
⊢
(∀𝑛 ∈
𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 176 | 174, 175 | sylib 218 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 177 | 176, 135 | sylnibr 329 |
. . . . . . . . 9
⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 178 | 177 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 179 | 140 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) |
| 180 | 178, 179 | neleqtrrd 2864 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ 𝐹) |
| 181 | 163, 180 | eldifd 3962 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) |
| 182 | 152, 181 | eqelssd 4005 |
. . . . 5
⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 183 | 182 | fveq2d 6910 |
. . . 4
⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
| 184 | 126, 183 | eqtr2d 2778 |
. . 3
⊢ (𝜑 → (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))) |
| 185 | 103, 184 | breq12d 5156 |
. 2
⊢ (𝜑 → (𝑆 ⇝ (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))) |
| 186 | 101, 185 | mpbird 257 |
1
⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩
𝑛 ∈ 𝑍 (𝐸‘𝑛))) |