| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | uc1pval.c | . 2
⊢ 𝐶 =
(Unic1p‘𝑅) | 
| 2 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) =
(Poly1‘𝑅)) | 
| 3 |  | uc1pval.p | . . . . . . . 8
⊢ 𝑃 = (Poly1‘𝑅) | 
| 4 | 2, 3 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃) | 
| 5 | 4 | fveq2d 6910 | . . . . . 6
⊢ (𝑟 = 𝑅 →
(Base‘(Poly1‘𝑟)) = (Base‘𝑃)) | 
| 6 |  | uc1pval.b | . . . . . 6
⊢ 𝐵 = (Base‘𝑃) | 
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5
⊢ (𝑟 = 𝑅 →
(Base‘(Poly1‘𝑟)) = 𝐵) | 
| 8 | 4 | fveq2d 6910 | . . . . . . . 8
⊢ (𝑟 = 𝑅 →
(0g‘(Poly1‘𝑟)) = (0g‘𝑃)) | 
| 9 |  | uc1pval.z | . . . . . . . 8
⊢  0 =
(0g‘𝑃) | 
| 10 | 8, 9 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑟 = 𝑅 →
(0g‘(Poly1‘𝑟)) = 0 ) | 
| 11 | 10 | neeq2d 3001 | . . . . . 6
⊢ (𝑟 = 𝑅 → (𝑓 ≠
(0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 )) | 
| 12 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅)) | 
| 13 |  | uc1pval.d | . . . . . . . . . 10
⊢ 𝐷 = (deg1‘𝑅) | 
| 14 | 12, 13 | eqtr4di 2795 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷) | 
| 15 | 14 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑓) = (𝐷‘𝑓)) | 
| 16 | 15 | fveq2d 6910 | . . . . . . 7
⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓))) | 
| 17 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | 
| 18 |  | uc1pval.u | . . . . . . . 8
⊢ 𝑈 = (Unit‘𝑅) | 
| 19 | 17, 18 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) | 
| 20 | 16, 19 | eleq12d 2835 | . . . . . 6
⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)) | 
| 21 | 11, 20 | anbi12d 632 | . . . . 5
⊢ (𝑟 = 𝑅 → ((𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟)) ↔ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈))) | 
| 22 | 7, 21 | rabeqbidv 3455 | . . . 4
⊢ (𝑟 = 𝑅 → {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))} = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}) | 
| 23 |  | df-uc1p 26171 | . . . 4
⊢
Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}) | 
| 24 | 6 | fvexi 6920 | . . . . 5
⊢ 𝐵 ∈ V | 
| 25 | 24 | rabex 5339 | . . . 4
⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ∈ V | 
| 26 | 22, 23, 25 | fvmpt 7016 | . . 3
⊢ (𝑅 ∈ V →
(Unic1p‘𝑅)
= {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}) | 
| 27 |  | fvprc 6898 | . . . 4
⊢ (¬
𝑅 ∈ V →
(Unic1p‘𝑅)
= ∅) | 
| 28 |  | ssrab2 4080 | . . . . . 6
⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ 𝐵 | 
| 29 |  | fvprc 6898 | . . . . . . . . . 10
⊢ (¬
𝑅 ∈ V →
(Poly1‘𝑅)
= ∅) | 
| 30 | 3, 29 | eqtrid 2789 | . . . . . . . . 9
⊢ (¬
𝑅 ∈ V → 𝑃 = ∅) | 
| 31 | 30 | fveq2d 6910 | . . . . . . . 8
⊢ (¬
𝑅 ∈ V →
(Base‘𝑃) =
(Base‘∅)) | 
| 32 |  | base0 17252 | . . . . . . . 8
⊢ ∅ =
(Base‘∅) | 
| 33 | 31, 32 | eqtr4di 2795 | . . . . . . 7
⊢ (¬
𝑅 ∈ V →
(Base‘𝑃) =
∅) | 
| 34 | 6, 33 | eqtrid 2789 | . . . . . 6
⊢ (¬
𝑅 ∈ V → 𝐵 = ∅) | 
| 35 | 28, 34 | sseqtrid 4026 | . . . . 5
⊢ (¬
𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ ∅) | 
| 36 |  | ss0 4402 | . . . . 5
⊢ ({𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ ∅ → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} = ∅) | 
| 37 | 35, 36 | syl 17 | . . . 4
⊢ (¬
𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} = ∅) | 
| 38 | 27, 37 | eqtr4d 2780 | . . 3
⊢ (¬
𝑅 ∈ V →
(Unic1p‘𝑅)
= {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}) | 
| 39 | 26, 38 | pm2.61i 182 | . 2
⊢
(Unic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} | 
| 40 | 1, 39 | eqtri 2765 | 1
⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧
((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} |