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Mirrors > Home > MPE Home > Th. List > mon1pn0 | Structured version Visualization version GIF version |
Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pn0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pn0.z | ⊢ 0 = (0g‘𝑃) |
mon1pn0.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
Ref | Expression |
---|---|
mon1pn0 | ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pn0.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
3 | uc1pn0.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
4 | eqid 2731 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | mon1pn0.m | . . 3 ⊢ 𝑀 = (Monic1p‘𝑅) | |
6 | eqid 2731 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | ismon1p 25589 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) = (1r‘𝑅))) |
8 | 7 | simp2bi 1146 | 1 ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ‘cfv 6532 Basecbs 17126 0gc0g 17367 1rcur 19963 Poly1cpl1 21630 coe1cco1 21631 deg1 cdg1 25498 Monic1pcmn1 25572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-1cn 11150 ax-addcl 11152 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-nn 12195 df-slot 17097 df-ndx 17109 df-base 17127 df-mon1 25577 |
This theorem is referenced by: mon1puc1p 25597 deg1submon1p 25599 irngnzply1 32593 mon1psubm 41719 |
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