| Step | Hyp | Ref
| Expression |
| 1 | | infnsuprnmpt.x |
. . . 4
⊢
Ⅎ𝑥𝜑 |
| 2 | | eqid 2737 |
. . . 4
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| 3 | | infnsuprnmpt.b |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 4 | 1, 2, 3 | rnmptssd 45201 |
. . 3
⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) |
| 5 | | infnsuprnmpt.a |
. . . 4
⊢ (𝜑 → 𝐴 ≠ ∅) |
| 6 | 1, 3, 2, 5 | rnmptn0 6264 |
. . 3
⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
| 7 | | infnsuprnmpt.l |
. . . 4
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) |
| 8 | 7 | rnmptlb 45250 |
. . 3
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) |
| 9 | | infrenegsup 12251 |
. . 3
⊢ ((ran
(𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < )) |
| 10 | 4, 6, 8, 9 | syl3anc 1373 |
. 2
⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < )) |
| 11 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵) |
| 12 | | rabidim2 45107 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 13 | 12 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 14 | | negex 11506 |
. . . . . . . . . . . 12
⊢ -𝑤 ∈ V |
| 15 | 2 | elrnmpt 5969 |
. . . . . . . . . . . 12
⊢ (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)) |
| 16 | 14, 15 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵) |
| 17 | 13, 16 | sylib 218 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵) |
| 18 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝑤 |
| 19 | 18 | nfneg 11504 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥-𝑤 |
| 20 | | nfmpt1 5250 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) |
| 21 | 20 | nfrn 5963 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥ran
(𝑥 ∈ 𝐴 ↦ 𝐵) |
| 22 | 19, 21 | nfel 2920 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
| 23 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥ℝ |
| 24 | 22, 23 | nfrabw 3475 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} |
| 25 | 18, 24 | nfel 2920 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} |
| 26 | 1, 25 | nfan 1899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) |
| 27 | | rabidim1 3459 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ℝ) |
| 28 | 27 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ℝ) |
| 29 | | negeq 11500 |
. . . . . . . . . . . . . . . 16
⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵) |
| 30 | 29 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤) |
| 31 | 30 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤) |
| 32 | | simp1r 1199 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ) |
| 33 | | recn 11245 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ ℝ → 𝑤 ∈
ℂ) |
| 34 | 33 | negnegd 11611 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤) |
| 35 | 32, 34 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤) |
| 36 | 31, 35 | eqtr2d 2778 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵) |
| 37 | 36 | 3exp 1120 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵))) |
| 38 | 28, 37 | syldan 591 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (𝑥 ∈ 𝐴 → (-𝑤 = 𝐵 → 𝑤 = -𝐵))) |
| 39 | 26, 38 | reximdai 3261 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)) |
| 40 | 17, 39 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) |
| 41 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) |
| 42 | 11, 40, 41 | elrnmptd 5974 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) |
| 43 | 42 | ex 412 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))) |
| 44 | | vex 3484 |
. . . . . . . . . . . . 13
⊢ 𝑤 ∈ V |
| 45 | 11 | elrnmpt 5969 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)) |
| 46 | 44, 45 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) |
| 47 | 46 | biimpi 216 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) |
| 48 | 47 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) |
| 49 | 18, 23 | nfel 2920 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥 𝑤 ∈ ℝ |
| 50 | 49, 22 | nfan 1899 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 51 | | simp3 1139 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 = -𝐵) |
| 52 | 3 | renegcld 11690 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ) |
| 53 | 52 | 3adant3 1133 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝐵 ∈ ℝ) |
| 54 | 51, 53 | eqeltrd 2841 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑤 ∈ ℝ) |
| 55 | | simp2 1138 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → 𝑥 ∈ 𝐴) |
| 56 | 51 | negeqd 11502 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵) |
| 57 | 3 | recnd 11289 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 58 | 57 | negnegd 11611 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵) |
| 59 | 58 | 3adant3 1133 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵) |
| 60 | 56, 59 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵) |
| 61 | | rspe 3249 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵) |
| 62 | 55, 60, 61 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵) |
| 63 | 14 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ V) |
| 64 | 2, 62, 63 | elrnmptd 5974 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 65 | 54, 64 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))) |
| 66 | 65 | 3exp 1120 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))))) |
| 67 | 1, 50, 66 | rexlimd 3266 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))) |
| 68 | 67 | imp 406 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))) |
| 69 | 48, 68 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))) |
| 70 | | rabid 3458 |
. . . . . . . . 9
⊢ (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))) |
| 71 | 69, 70 | sylibr 234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}) |
| 72 | 71 | ex 412 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})) |
| 73 | 43, 72 | impbid 212 |
. . . . . 6
⊢ (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))) |
| 74 | 73 | alrimiv 1927 |
. . . . 5
⊢ (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))) |
| 75 | | nfrab1 3457 |
. . . . . 6
⊢
Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} |
| 76 | | nfcv 2905 |
. . . . . 6
⊢
Ⅎ𝑤ran
(𝑥 ∈ 𝐴 ↦ -𝐵) |
| 77 | 75, 76 | cleqf 2934 |
. . . . 5
⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))) |
| 78 | 74, 77 | sylibr 234 |
. . . 4
⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵)) |
| 79 | 78 | supeq1d 9486 |
. . 3
⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) |
| 80 | 79 | negeqd 11502 |
. 2
⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) |
| 81 | | eqidd 2738 |
. 2
⊢ (𝜑 → -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) |
| 82 | 10, 80, 81 | 3eqtrd 2781 |
1
⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) |