Proof of Theorem limsupequzlem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | limsupequzlem.1 | . . . . 5
⊢
Ⅎ𝑘𝜑 | 
| 2 |  | eqid 2736 | . . . . . . 7
⊢
(ℤ≥‘𝐾) = (ℤ≥‘𝐾) | 
| 3 |  | limsupequzlem.7 | . . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ ℤ) | 
| 4 | 3 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝐾 ∈
ℤ) | 
| 5 |  | eluzelz 12889 | . . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) → 𝑘 ∈
ℤ) | 
| 6 | 5 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝑘 ∈
ℤ) | 
| 7 | 3 | zred 12724 | . . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ ℝ) | 
| 8 | 7 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝐾 ∈
ℝ) | 
| 9 | 8 | rexrd 11312 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝐾 ∈
ℝ*) | 
| 10 |  | zssxr 45413 | . . . . . . . . . 10
⊢ ℤ
⊆ ℝ* | 
| 11 |  | limsupequzlem.2 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 12 |  | limsupequzlem.5 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 13 |  | tpssi 4837 | . . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → {𝑀, 𝑁, 𝐾} ⊆ ℤ) | 
| 14 | 11, 12, 3, 13 | syl3anc 1372 | . . . . . . . . . . 11
⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℤ) | 
| 15 |  | xrltso 13184 | . . . . . . . . . . . . 13
⊢  < Or
ℝ* | 
| 16 | 15 | a1i 11 | . . . . . . . . . . . 12
⊢ (𝜑 → < Or
ℝ*) | 
| 17 |  | tpfi 9366 | . . . . . . . . . . . . 13
⊢ {𝑀, 𝑁, 𝐾} ∈ Fin | 
| 18 | 17 | a1i 11 | . . . . . . . . . . . 12
⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ∈ Fin) | 
| 19 | 11 | tpnzd 4779 | . . . . . . . . . . . 12
⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ≠ ∅) | 
| 20 | 10 | a1i 11 | . . . . . . . . . . . . 13
⊢ (𝜑 → ℤ ⊆
ℝ*) | 
| 21 | 14, 20 | sstrd 3993 | . . . . . . . . . . . 12
⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆
ℝ*) | 
| 22 |  | fisupcl 9510 | . . . . . . . . . . . 12
⊢ (( <
Or ℝ* ∧ ({𝑀, 𝑁, 𝐾} ∈ Fin ∧ {𝑀, 𝑁, 𝐾} ≠ ∅ ∧ {𝑀, 𝑁, 𝐾} ⊆ ℝ*)) →
sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈ {𝑀, 𝑁, 𝐾}) | 
| 23 | 16, 18, 19, 21, 22 | syl13anc 1373 | . . . . . . . . . . 11
⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈ {𝑀, 𝑁, 𝐾}) | 
| 24 | 14, 23 | sseldd 3983 | . . . . . . . . . 10
⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
ℤ) | 
| 25 | 10, 24 | sselid 3980 | . . . . . . . . 9
⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
ℝ*) | 
| 26 | 25 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
ℝ*) | 
| 27 |  | eluzelre 12890 | . . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) → 𝑘 ∈
ℝ) | 
| 28 | 27 | adantl 481 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝑘 ∈
ℝ) | 
| 29 | 28 | rexrd 11312 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝑘 ∈
ℝ*) | 
| 30 |  | tpid3g 4771 | . . . . . . . . . . 11
⊢ (𝐾 ∈ ℤ → 𝐾 ∈ {𝑀, 𝑁, 𝐾}) | 
| 31 | 3, 30 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ {𝑀, 𝑁, 𝐾}) | 
| 32 |  | eqid 2736 | . . . . . . . . . 10
⊢
sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) = sup({𝑀, 𝑁, 𝐾}, ℝ*, <
) | 
| 33 | 21, 31, 32 | supxrubd 45123 | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)) | 
| 34 | 33 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝐾 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)) | 
| 35 |  | eluzle 12892 | . . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) →
sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ≤ 𝑘) | 
| 36 | 35 | adantl 481 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ≤ 𝑘) | 
| 37 | 9, 26, 29, 34, 36 | xrletrd 13205 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝐾 ≤ 𝑘) | 
| 38 | 2, 4, 6, 37 | eluzd 45425 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
𝑘 ∈
(ℤ≥‘𝐾)) | 
| 39 |  | limsupequzlem.8 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | 
| 40 | 38, 39 | syldan 591 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) →
(𝐹‘𝑘) = (𝐺‘𝑘)) | 
| 41 | 1, 40 | ralrimia 3257 | . . . 4
⊢ (𝜑 → ∀𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))(𝐹‘𝑘) = (𝐺‘𝑘)) | 
| 42 |  | limsupequzlem.4 | . . . . 5
⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) | 
| 43 |  | limsupequzlem.6 | . . . . 5
⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) | 
| 44 |  | eqid 2736 | . . . . . . 7
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) | 
| 45 |  | tpid1g 4768 | . . . . . . . . 9
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ {𝑀, 𝑁, 𝐾}) | 
| 46 | 11, 45 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ {𝑀, 𝑁, 𝐾}) | 
| 47 | 21, 46, 32 | supxrubd 45123 | . . . . . . 7
⊢ (𝜑 → 𝑀 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)) | 
| 48 | 44, 11, 24, 47 | eluzd 45425 | . . . . . 6
⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
(ℤ≥‘𝑀)) | 
| 49 |  | uzss 12902 | . . . . . 6
⊢
(sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
(ℤ≥‘𝑀) →
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑀)) | 
| 50 | 48, 49 | syl 17 | . . . . 5
⊢ (𝜑 →
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑀)) | 
| 51 |  | eqid 2736 | . . . . . . 7
⊢
(ℤ≥‘𝑁) = (ℤ≥‘𝑁) | 
| 52 |  | tpid2g 4770 | . . . . . . . . 9
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ {𝑀, 𝑁, 𝐾}) | 
| 53 | 12, 52 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ {𝑀, 𝑁, 𝐾}) | 
| 54 | 21, 53, 32 | supxrubd 45123 | . . . . . . 7
⊢ (𝜑 → 𝑁 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)) | 
| 55 | 51, 12, 24, 54 | eluzd 45425 | . . . . . 6
⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
(ℤ≥‘𝑁)) | 
| 56 |  | uzss 12902 | . . . . . 6
⊢
(sup({𝑀, 𝑁, 𝐾}, ℝ*, < ) ∈
(ℤ≥‘𝑁) →
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑁)) | 
| 57 | 55, 56 | syl 17 | . . . . 5
⊢ (𝜑 →
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑁)) | 
| 58 |  | fvreseq0 7057 | . . . . 5
⊢ (((𝐹 Fn
(ℤ≥‘𝑀) ∧ 𝐺 Fn (ℤ≥‘𝑁)) ∧
((ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑀) ∧
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) ⊆
(ℤ≥‘𝑁))) → ((𝐹 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) = (𝐺 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) ↔
∀𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))(𝐹‘𝑘) = (𝐺‘𝑘))) | 
| 59 | 42, 43, 50, 57, 58 | syl22anc 838 | . . . 4
⊢ (𝜑 → ((𝐹 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) = (𝐺 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) ↔
∀𝑘 ∈
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))(𝐹‘𝑘) = (𝐺‘𝑘))) | 
| 60 | 41, 59 | mpbird 257 | . . 3
⊢ (𝜑 → (𝐹 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < ))) = (𝐺 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)))) | 
| 61 | 60 | fveq2d 6909 | . 2
⊢ (𝜑 → (lim sup‘(𝐹 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )))) = (lim
sup‘(𝐺 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, <
))))) | 
| 62 |  | eqid 2736 | . . 3
⊢
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )) =
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, <
)) | 
| 63 |  | fvexd 6920 | . . . 4
⊢ (𝜑 →
(ℤ≥‘𝑀) ∈ V) | 
| 64 | 42, 63 | fnexd 7239 | . . 3
⊢ (𝜑 → 𝐹 ∈ V) | 
| 65 | 42 | fndmd 6672 | . . . 4
⊢ (𝜑 → dom 𝐹 = (ℤ≥‘𝑀)) | 
| 66 |  | uzssz 12900 | . . . 4
⊢
(ℤ≥‘𝑀) ⊆ ℤ | 
| 67 | 65, 66 | eqsstrdi 4027 | . . 3
⊢ (𝜑 → dom 𝐹 ⊆ ℤ) | 
| 68 | 24, 62, 64, 67 | limsupresuz2 45729 | . 2
⊢ (𝜑 → (lim sup‘(𝐹 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )))) = (lim
sup‘𝐹)) | 
| 69 |  | fvexd 6920 | . . . 4
⊢ (𝜑 →
(ℤ≥‘𝑁) ∈ V) | 
| 70 | 43, 69 | fnexd 7239 | . . 3
⊢ (𝜑 → 𝐺 ∈ V) | 
| 71 | 43 | fndmd 6672 | . . . 4
⊢ (𝜑 → dom 𝐺 = (ℤ≥‘𝑁)) | 
| 72 |  | uzssz 12900 | . . . 4
⊢
(ℤ≥‘𝑁) ⊆ ℤ | 
| 73 | 71, 72 | eqsstrdi 4027 | . . 3
⊢ (𝜑 → dom 𝐺 ⊆ ℤ) | 
| 74 | 24, 62, 70, 73 | limsupresuz2 45729 | . 2
⊢ (𝜑 → (lim sup‘(𝐺 ↾
(ℤ≥‘sup({𝑀, 𝑁, 𝐾}, ℝ*, < )))) = (lim
sup‘𝐺)) | 
| 75 | 61, 68, 74 | 3eqtr3d 2784 | 1
⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |