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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 3078 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
2 | fveq2 6670 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
3 | fveq2 6670 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
4 | 2, 3 | fveq12d 6677 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
5 | 4 | eleq1d 2897 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
6 | 1, 5 | anbi12d 632 | . . 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 24733 | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} |
14 | 6, 13 | elrab2 3683 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) |
15 | 3anass 1091 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))) | |
16 | 14, 15 | bitr4i 280 | 1 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ‘cfv 6355 Basecbs 16483 0gc0g 16713 Unitcui 19389 Poly1cpl1 20345 coe1cco1 20346 deg1 cdg1 24648 Unic1pcuc1p 24720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-slot 16487 df-base 16489 df-uc1p 24725 |
This theorem is referenced by: uc1pcl 24737 uc1pn0 24739 uc1pldg 24742 mon1puc1p 24744 drnguc1p 24764 |
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