Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2996 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
2 | fveq2 6668 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
3 | fveq2 6668 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
4 | 2, 3 | fveq12d 6675 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
5 | 4 | eleq1d 2817 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
6 | 1, 5 | anbi12d 634 | . . 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 24884 | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} |
14 | 6, 13 | elrab2 3588 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) |
15 | 3anass 1096 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) | |
16 | 14, 15 | bitr4i 281 | 1 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ‘cfv 6333 Basecbs 16579 0gc0g 16809 Unitcui 19504 Poly1cpl1 20945 coe1cco1 20946 deg1 cdg1 24796 Unic1pcuc1p 24871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fv 6341 df-slot 16583 df-base 16585 df-uc1p 24876 |
This theorem is referenced by: uc1pcl 24888 uc1pn0 24890 uc1pldg 24893 mon1puc1p 24895 drnguc1p 24915 |
Copyright terms: Public domain | W3C validator |