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Mirrors > Home > MPE Home > Th. List > isuc1p | Structured version Visualization version GIF version |
Description: Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pval.b | ⊢ 𝐵 = (Base‘𝑃) |
uc1pval.z | ⊢ 0 = (0g‘𝑃) |
uc1pval.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
uc1pval.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
uc1pval.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
isuc1p | ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 2856 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
2 | fveq2 6191 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
4 | 2, 3 | fveq12d 6197 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
5 | 4 | eleq1d 2686 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
6 | 1, 5 | anbi12d 747 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈) ↔ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) |
7 | uc1pval.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | uc1pval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | uc1pval.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
10 | uc1pval.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
11 | uc1pval.c | . . . 4 ⊢ 𝐶 = (Unic1p‘𝑅) | |
12 | uc1pval.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
13 | 7, 8, 9, 10, 11, 12 | uc1pval 23899 | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} |
14 | 6, 13 | elrab2 3366 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) |
15 | 3anass 1042 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) | |
16 | 14, 15 | bitr4i 267 | 1 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 Basecbs 15857 0gc0g 16100 Unitcui 18639 Poly1cpl1 19547 coe1cco1 19548 deg1 cdg1 23814 Unic1pcuc1p 23886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-slot 15861 df-base 15863 df-uc1p 23891 |
This theorem is referenced by: uc1pcl 23903 uc1pn0 23905 uc1pldg 23908 mon1puc1p 23910 drnguc1p 23930 |
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