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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 3075 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
2 | fveq2 6663 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
3 | fveq2 6663 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
4 | 2, 3 | fveq12d 6670 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
5 | 4 | eqeq1d 2820 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
6 | 1, 5 | anbi12d 630 | . . 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 24662 | . . 3 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} |
14 | 6, 13 | elrab2 3680 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) |
15 | 3anass 1087 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) | |
16 | 14, 15 | bitr4i 279 | 1 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 Basecbs 16471 0gc0g 16701 1rcur 19180 Poly1cpl1 20273 coe1cco1 20274 deg1 cdg1 24575 Monic1pcmn1 24646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-slot 16475 df-base 16477 df-mon1 24651 |
This theorem is referenced by: mon1pcl 24665 mon1pn0 24667 mon1pldg 24670 uc1pmon1p 24672 ply1remlem 24683 mon1pid 39683 mon1psubm 39684 |
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