| Step | Hyp | Ref
| Expression |
| 1 | | pimincfltioc.y |
. . . . . . 7
⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} |
| 2 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴 |
| 3 | 1, 2 | eqsstri 4030 |
. . . . . 6
⊢ 𝑌 ⊆ 𝐴 |
| 4 | 3 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
| 5 | | pimincfltioc.a |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 6 | 4, 5 | sstrd 3994 |
. . . 4
⊢ (𝜑 → 𝑌 ⊆ ℝ) |
| 7 | | pimincfltioc.c |
. . . 4
⊢ 𝑆 = sup(𝑌, ℝ*, <
) |
| 8 | | pimincfltioc.e |
. . . 4
⊢ (𝜑 → 𝑆 ∈ 𝑌) |
| 9 | | pimincfltioc.d |
. . . 4
⊢ 𝐼 = (-∞(,]𝑆) |
| 10 | 6, 7, 8, 9 | ressiocsup 45567 |
. . 3
⊢ (𝜑 → 𝑌 ⊆ 𝐼) |
| 11 | 10, 4 | ssind 4241 |
. 2
⊢ (𝜑 → 𝑌 ⊆ (𝐼 ∩ 𝐴)) |
| 12 | | pimincfltioc.x |
. . . 4
⊢
Ⅎ𝑥𝜑 |
| 13 | | elinel2 4202 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐼 ∩ 𝐴) → 𝑥 ∈ 𝐴) |
| 14 | 13 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ 𝐴) |
| 15 | | pimincfltioc.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
| 16 | 15 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → 𝐹:𝐴⟶ℝ*) |
| 17 | 16, 14 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑥) ∈
ℝ*) |
| 18 | 3, 8 | sselid 3981 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 19 | 15, 18 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑆) ∈
ℝ*) |
| 20 | 19 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑆) ∈
ℝ*) |
| 21 | | pimincfltioc.r |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈
ℝ*) |
| 22 | 21 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → 𝑅 ∈
ℝ*) |
| 23 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → (𝑧 ∈ (𝐼 ∩ 𝐴) ↔ 𝑥 ∈ (𝐼 ∩ 𝐴))) |
| 24 | 23 | anbi2d 630 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) ↔ (𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)))) |
| 25 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) |
| 26 | 25 | breq1d 5153 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ≤ (𝐹‘𝑆) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑆))) |
| 27 | 24, 26 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → (((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑧) ≤ (𝐹‘𝑆)) ↔ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑥) ≤ (𝐹‘𝑆)))) |
| 28 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑧 ∈ (𝐼 ∩ 𝐴) |
| 29 | 12, 28 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) |
| 30 | | pimincfltioc.h |
. . . . . . . . . . 11
⊢
Ⅎ𝑦𝜑 |
| 31 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦 𝑧 ∈ (𝐼 ∩ 𝐴) |
| 32 | 30, 31 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) |
| 33 | | pimincfltioc.i |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 34 | 33 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 35 | | elinel2 4202 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝐼 ∩ 𝐴) → 𝑧 ∈ 𝐴) |
| 36 | 35 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → 𝑧 ∈ 𝐴) |
| 37 | 18 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → 𝑆 ∈ 𝐴) |
| 38 | | mnfxr 11318 |
. . . . . . . . . . . 12
⊢ -∞
∈ ℝ* |
| 39 | 38 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → -∞ ∈
ℝ*) |
| 40 | | ressxr 11305 |
. . . . . . . . . . . . 13
⊢ ℝ
⊆ ℝ* |
| 41 | 6, 8 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ ℝ) |
| 42 | 40, 41 | sselid 3981 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈
ℝ*) |
| 43 | 42 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → 𝑆 ∈
ℝ*) |
| 44 | | elinel1 4201 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (𝐼 ∩ 𝐴) → 𝑧 ∈ 𝐼) |
| 45 | 44, 9 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝐼 ∩ 𝐴) → 𝑧 ∈ (-∞(,]𝑆)) |
| 46 | 45 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → 𝑧 ∈ (-∞(,]𝑆)) |
| 47 | | iocleub 45516 |
. . . . . . . . . . 11
⊢
((-∞ ∈ ℝ* ∧ 𝑆 ∈ ℝ* ∧ 𝑧 ∈ (-∞(,]𝑆)) → 𝑧 ≤ 𝑆) |
| 48 | 39, 43, 46, 47 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → 𝑧 ≤ 𝑆) |
| 49 | 29, 32, 34, 36, 37, 48 | dmrelrnrel 45231 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑧) ≤ (𝐹‘𝑆)) |
| 50 | 27, 49 | chvarvv 1998 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑥) ≤ (𝐹‘𝑆)) |
| 51 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑆 → (𝐹‘𝑥) = (𝐹‘𝑆)) |
| 52 | 51 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑆 → ((𝐹‘𝑥) < 𝑅 ↔ (𝐹‘𝑆) < 𝑅)) |
| 53 | 52, 1 | elrab2 3695 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ 𝑌 ↔ (𝑆 ∈ 𝐴 ∧ (𝐹‘𝑆) < 𝑅)) |
| 54 | 8, 53 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 ∈ 𝐴 ∧ (𝐹‘𝑆) < 𝑅)) |
| 55 | 54 | simprd 495 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑆) < 𝑅) |
| 56 | 55 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑆) < 𝑅) |
| 57 | 17, 20, 22, 50, 56 | xrlelttrd 13202 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝐹‘𝑥) < 𝑅) |
| 58 | 14, 57 | jca 511 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝑅)) |
| 59 | 1 | reqabi 3460 |
. . . . . 6
⊢ (𝑥 ∈ 𝑌 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝑅)) |
| 60 | 58, 59 | sylibr 234 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ 𝑌) |
| 61 | 60 | ex 412 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐼 ∩ 𝐴) → 𝑥 ∈ 𝑌)) |
| 62 | 12, 61 | ralrimi 3257 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ (𝐼 ∩ 𝐴)𝑥 ∈ 𝑌) |
| 63 | 28 | nfci 2893 |
. . . 4
⊢
Ⅎ𝑥(𝐼 ∩ 𝐴) |
| 64 | | nfrab1 3457 |
. . . . 5
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} |
| 65 | 1, 64 | nfcxfr 2903 |
. . . 4
⊢
Ⅎ𝑥𝑌 |
| 66 | 63, 65 | dfss3f 3975 |
. . 3
⊢ ((𝐼 ∩ 𝐴) ⊆ 𝑌 ↔ ∀𝑥 ∈ (𝐼 ∩ 𝐴)𝑥 ∈ 𝑌) |
| 67 | 62, 66 | sylibr 234 |
. 2
⊢ (𝜑 → (𝐼 ∩ 𝐴) ⊆ 𝑌) |
| 68 | 11, 67 | eqssd 4001 |
1
⊢ (𝜑 → 𝑌 = (𝐼 ∩ 𝐴)) |