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| Mirrors > Home > MPE Home > Th. List > ismon1p | Structured version Visualization version GIF version | ||
| Description: Being a monic 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) |
| mon1pval.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
| mon1pval.o | ⊢ 1 = (1r‘𝑅) |
| Ref | Expression |
|---|---|
| ismon1p | ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neeq1 3003 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
| 4 | 2, 3 | fveq12d 6913 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
| 5 | 4 | eqeq1d 2739 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
| 6 | 1, 5 | anbi12d 632 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ) ↔ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) |
| 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 | mon1pval.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
| 12 | mon1pval.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
| 13 | 7, 8, 9, 10, 11, 12 | mon1pval 26181 | . . 3 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} |
| 14 | 6, 13 | elrab2 3695 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) |
| 15 | 3anass 1095 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) | |
| 16 | 14, 15 | bitr4i 278 | 1 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 Basecbs 17247 0gc0g 17484 1rcur 20178 Poly1cpl1 22178 coe1cco1 22179 deg1cdg1 26093 Monic1pcmn1 26165 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-mon1 26170 |
| This theorem is referenced by: mon1pcl 26184 mon1pn0 26186 mon1pldg 26189 uc1pmon1p 26191 mon1pid 26193 ply1remlem 26204 0ringmon1p 33583 ressply1mon1p 33593 rtelextdg2lem 33767 2sqr3minply 33791 mon1psubm 43211 |
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