| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-rprm 20433 | . 2
⊢ RPrime =
(𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))}) | 
| 2 |  | fvexd 6921 | . . 3
⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V) | 
| 3 |  | simpr 484 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟)) | 
| 4 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | 
| 5 | 4 | adantr 480 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅)) | 
| 6 | 3, 5 | eqtrd 2777 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅)) | 
| 7 |  | rprmval.b | . . . . . 6
⊢ 𝐵 = (Base‘𝑅) | 
| 8 | 6, 7 | eqtr4di 2795 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵) | 
| 9 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | 
| 10 |  | rprmval.u | . . . . . . . 8
⊢ 𝑈 = (Unit‘𝑅) | 
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) | 
| 12 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | 
| 13 |  | rprmval.1 | . . . . . . . . 9
⊢  0 =
(0g‘𝑅) | 
| 14 | 12, 13 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) | 
| 15 | 14 | sneqd 4638 | . . . . . . 7
⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 }) | 
| 16 | 11, 15 | uneq12d 4169 | . . . . . 6
⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 })) | 
| 17 | 16 | adantr 480 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 })) | 
| 18 | 8, 17 | difeq12d 4127 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 }))) | 
| 19 |  | fvexd 6921 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∥r‘𝑟) ∈ V) | 
| 20 |  | eqidd 2738 | . . . . . . . . 9
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑝 = 𝑝) | 
| 21 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑟)) | 
| 22 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑟 = 𝑅 → (∥r‘𝑟) =
(∥r‘𝑅)) | 
| 23 | 22 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) →
(∥r‘𝑟) = (∥r‘𝑅)) | 
| 24 | 21, 23 | eqtrd 2777 | . . . . . . . . . 10
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑅)) | 
| 25 |  | rprmval.d | . . . . . . . . . 10
⊢  ∥ =
(∥r‘𝑅) | 
| 26 | 24, 25 | eqtr4di 2795 | . . . . . . . . 9
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = ∥ ) | 
| 27 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | 
| 28 |  | rprmval.m | . . . . . . . . . . . 12
⊢  · =
(.r‘𝑅) | 
| 29 | 27, 28 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) | 
| 30 | 29 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) →
(.r‘𝑟) =
·
) | 
| 31 | 30 | oveqd 7448 | . . . . . . . . 9
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦)) | 
| 32 | 20, 26, 31 | breq123d 5157 | . . . . . . . 8
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑(𝑥(.r‘𝑟)𝑦) ↔ 𝑝 ∥ (𝑥 · 𝑦))) | 
| 33 | 26 | breqd 5154 | . . . . . . . . 9
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑥 ↔ 𝑝 ∥ 𝑥)) | 
| 34 | 26 | breqd 5154 | . . . . . . . . 9
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑦 ↔ 𝑝 ∥ 𝑦)) | 
| 35 | 33, 34 | orbi12d 919 | . . . . . . . 8
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → ((𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦) ↔ (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))) | 
| 36 | 32, 35 | imbi12d 344 | . . . . . . 7
⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → ((𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))) | 
| 37 | 19, 36 | sbcied 3832 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) →
([(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))) | 
| 38 | 8, 37 | raleqbidv 3346 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))) | 
| 39 | 8, 38 | raleqbidv 3346 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))) | 
| 40 | 18, 39 | rabeqbidv 3455 | . . 3
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → {𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}) | 
| 41 | 2, 40 | csbied 3935 | . 2
⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}) | 
| 42 |  | elex 3501 | . 2
⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | 
| 43 | 7 | fvexi 6920 | . . . . 5
⊢ 𝐵 ∈ V | 
| 44 | 43 | difexi 5330 | . . . 4
⊢ (𝐵 ∖ (𝑈 ∪ { 0 })) ∈
V | 
| 45 | 44 | rabex 5339 | . . 3
⊢ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))} ∈ V | 
| 46 | 45 | a1i 11 | . 2
⊢ (𝑅 ∈ 𝑉 → {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))} ∈ V) | 
| 47 | 1, 41, 42, 46 | fvmptd3 7039 | 1
⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}) |