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Mirrors > Home > MPE Home > Th. List > mon1pldg | Structured version Visualization version GIF version |
Description: Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
mon1pldg.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
mon1pldg.o | ⊢ 1 = (1r‘𝑅) |
mon1pldg.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
Ref | Expression |
---|---|
mon1pldg | ⊢ (𝐹 ∈ 𝑀 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
3 | eqid 2738 | . . 3 ⊢ (0g‘(Poly1‘𝑅)) = (0g‘(Poly1‘𝑅)) | |
4 | mon1pldg.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
5 | mon1pldg.m | . . 3 ⊢ 𝑀 = (Monic1p‘𝑅) | |
6 | mon1pldg.o | . . 3 ⊢ 1 = (1r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | ismon1p 25212 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
8 | 7 | simp3bi 1145 | 1 ⊢ (𝐹 ∈ 𝑀 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ‘cfv 6418 Basecbs 16840 0gc0g 17067 1rcur 19652 Poly1cpl1 21258 coe1cco1 21259 deg1 cdg1 25121 Monic1pcmn1 25195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-1cn 10860 ax-addcl 10862 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-nn 11904 df-slot 16811 df-ndx 16823 df-base 16841 df-mon1 25200 |
This theorem is referenced by: mon1puc1p 25220 deg1submon1p 25222 mon1psubm 40947 |
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