| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mbfinf.2 | . . 3
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) | 
| 2 |  | mbfinf.5 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) | 
| 3 | 2 | anass1rs 655 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ) | 
| 4 | 3 | fmpttd 7134 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ) | 
| 5 | 4 | frnd 6743 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ) | 
| 6 |  | mbfinf.3 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 7 |  | uzid 12894 | . . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) | 
| 8 | 6, 7 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | 
| 9 |  | mbfinf.1 | . . . . . . . . . . 11
⊢ 𝑍 =
(ℤ≥‘𝑀) | 
| 10 | 8, 9 | eleqtrrdi 2851 | . . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ 𝑍) | 
| 11 | 10 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍) | 
| 12 |  | eqid 2736 | . . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵) | 
| 13 | 12, 3 | dmmptd 6712 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍) | 
| 14 | 11, 13 | eleqtrrd 2843 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵)) | 
| 15 | 14 | ne0d 4341 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅) | 
| 16 |  | dm0rn0 5934 | . . . . . . . 8
⊢ (dom
(𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅) | 
| 17 | 16 | necon3bii 2992 | . . . . . . 7
⊢ (dom
(𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅) | 
| 18 | 15, 17 | sylib 218 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅) | 
| 19 |  | mbfinf.6 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵) | 
| 20 | 4 | ffnd 6736 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍) | 
| 21 |  | breq2 5146 | . . . . . . . . . . 11
⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚))) | 
| 22 | 21 | ralrn 7107 | . . . . . . . . . 10
⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚))) | 
| 23 | 20, 22 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚))) | 
| 24 |  | nfcv 2904 | . . . . . . . . . . . 12
⊢
Ⅎ𝑛𝑦 | 
| 25 |  | nfcv 2904 | . . . . . . . . . . . 12
⊢
Ⅎ𝑛
≤ | 
| 26 |  | nffvmpt1 6916 | . . . . . . . . . . . 12
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) | 
| 27 | 24, 25, 26 | nfbr 5189 | . . . . . . . . . . 11
⊢
Ⅎ𝑛 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) | 
| 28 |  | nfv 1913 | . . . . . . . . . . 11
⊢
Ⅎ𝑚 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) | 
| 29 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) | 
| 30 | 29 | breq2d 5154 | . . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) | 
| 31 | 27, 28, 30 | cbvralw 3305 | . . . . . . . . . 10
⊢
(∀𝑚 ∈
𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) | 
| 32 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | 
| 33 | 12 | fvmpt2 7026 | . . . . . . . . . . . . 13
⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵) | 
| 34 | 32, 3, 33 | syl2anc 584 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵) | 
| 35 | 34 | breq2d 5154 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑦 ≤ 𝐵)) | 
| 36 | 35 | ralbidva 3175 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) | 
| 37 | 31, 36 | bitrid 283 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) | 
| 38 | 23, 37 | bitrd 279 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) | 
| 39 | 38 | rexbidv 3178 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) | 
| 40 | 19, 39 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧) | 
| 41 |  | infrenegsup 12252 | . . . . . 6
⊢ ((ran
(𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < )) | 
| 42 | 5, 18, 40, 41 | syl3anc 1372 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < )) | 
| 43 |  | rabid 3457 | . . . . . . . . . 10
⊢ (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ (𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵))) | 
| 44 | 3 | recnd 11290 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ) | 
| 45 | 44 | adantlr 715 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ) | 
| 46 |  | simplr 768 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℝ) | 
| 47 | 46 | recnd 11290 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℂ) | 
| 48 |  | negcon2 11563 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵)) | 
| 49 | 45, 47, 48 | syl2anc 584 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵)) | 
| 50 |  | eqcom 2743 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = -𝐵 ↔ -𝐵 = 𝑟) | 
| 51 | 49, 50 | bitrdi 287 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ -𝐵 = 𝑟)) | 
| 52 | 34 | adantlr 715 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵) | 
| 53 | 52 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ 𝐵 = -𝑟)) | 
| 54 |  | negex 11507 | . . . . . . . . . . . . . . . . . . . . 21
⊢ -𝐵 ∈ V | 
| 55 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ 𝑍 ↦ -𝐵) = (𝑛 ∈ 𝑍 ↦ -𝐵) | 
| 56 | 55 | fvmpt2 7026 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ 𝑍 ∧ -𝐵 ∈ V) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵) | 
| 57 | 54, 56 | mpan2 691 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵) | 
| 58 | 57 | adantl 481 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵) | 
| 59 | 58 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟 ↔ -𝐵 = 𝑟)) | 
| 60 | 51, 53, 59 | 3bitr4d 311 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)) | 
| 61 | 60 | ralrimiva 3145 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)) | 
| 62 | 26 | nfeq1 2920 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 | 
| 63 |  | nffvmpt1 6916 | . . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) | 
| 64 | 63 | nfeq1 2920 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟 | 
| 65 | 62, 64 | nfbi 1902 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑛(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟) | 
| 66 |  | nfv 1913 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑚(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟) | 
| 67 |  | fveqeq2 6914 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟)) | 
| 68 |  | fveqeq2 6914 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)) | 
| 69 | 67, 68 | bibi12d 345 | . . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑛 → ((((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟) ↔ (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))) | 
| 70 | 65, 66, 69 | cbvralw 3305 | . . . . . . . . . . . . . . . 16
⊢
(∀𝑚 ∈
𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟) ↔ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)) | 
| 71 | 61, 70 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑚 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)) | 
| 72 | 71 | r19.21bi 3250 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)) | 
| 73 | 72 | rexbidva 3176 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)) | 
| 74 | 20 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍) | 
| 75 |  | fvelrnb 6968 | . . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟)) | 
| 76 | 74, 75 | syl 17 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟)) | 
| 77 | 3 | renegcld 11691 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → -𝐵 ∈ ℝ) | 
| 78 | 77 | fmpttd 7134 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ) | 
| 79 | 78 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ) | 
| 80 | 79 | ffnd 6736 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍) | 
| 81 |  | fvelrnb 6968 | . . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍 → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)) | 
| 82 | 80, 81 | syl 17 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)) | 
| 83 | 73, 76, 82 | 3bitr4d 311 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))) | 
| 84 | 83 | pm5.32da 579 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))) | 
| 85 | 78 | frnd 6743 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ -𝐵) ⊆ ℝ) | 
| 86 | 85 | sseld 3981 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) → 𝑟 ∈ ℝ)) | 
| 87 | 86 | pm4.71rd 562 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))) | 
| 88 | 84, 87 | bitr4d 282 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))) | 
| 89 | 43, 88 | bitrid 283 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))) | 
| 90 | 89 | alrimiv 1926 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))) | 
| 91 |  | nfrab1 3456 | . . . . . . . . 9
⊢
Ⅎ𝑟{𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} | 
| 92 |  | nfcv 2904 | . . . . . . . . 9
⊢
Ⅎ𝑟ran
(𝑛 ∈ 𝑍 ↦ -𝐵) | 
| 93 | 91, 92 | cleqf 2933 | . . . . . . . 8
⊢ ({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))) | 
| 94 | 90, 93 | sylibr 234 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵)) | 
| 95 | 94 | supeq1d 9487 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) | 
| 96 | 95 | negeqd 11503 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) | 
| 97 | 42, 96 | eqtrd 2776 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) | 
| 98 | 97 | mpteq2dva 5241 | . . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))) | 
| 99 | 1, 98 | eqtrid 2788 | . 2
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))) | 
| 100 |  | ltso 11342 | . . . . 5
⊢  < Or
ℝ | 
| 101 | 100 | supex 9504 | . . . 4
⊢ sup(ran
(𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ) ∈ V | 
| 102 | 101 | a1i 11 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ) ∈
V) | 
| 103 |  | eqid 2736 | . . . 4
⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) | 
| 104 | 2 | anassrs 467 | . . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | 
| 105 |  | mbfinf.4 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | 
| 106 | 104, 105 | mbfneg 25686 | . . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn) | 
| 107 | 2 | renegcld 11691 | . . . 4
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → -𝐵 ∈ ℝ) | 
| 108 |  | renegcl 11573 | . . . . . . 7
⊢ (𝑦 ∈ ℝ → -𝑦 ∈
ℝ) | 
| 109 | 108 | ad2antrl 728 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → -𝑦 ∈ ℝ) | 
| 110 |  | simplr 768 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ) | 
| 111 | 3 | adantlr 715 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ) | 
| 112 | 110, 111 | lenegd 11843 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦)) | 
| 113 | 112 | ralbidva 3175 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦)) | 
| 114 | 113 | biimpd 229 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦)) | 
| 115 | 114 | impr 454 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) | 
| 116 |  | brralrspcev 5202 | . . . . . 6
⊢ ((-𝑦 ∈ ℝ ∧
∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧) | 
| 117 | 109, 115,
116 | syl2anc 584 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧) | 
| 118 | 19, 117 | rexlimddv 3160 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧) | 
| 119 | 9, 103, 6, 106, 107, 118 | mbfsup 25700 | . . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈
MblFn) | 
| 120 | 102, 119 | mbfneg 25686 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈
MblFn) | 
| 121 | 99, 120 | eqeltrd 2840 | 1
⊢ (𝜑 → 𝐺 ∈ MblFn) |