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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 2824 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
2 | eqid 2824 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
3 | eqid 2824 | . . 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 24739 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
8 | 7 | simp3bi 1143 | 1 ⊢ (𝐹 ∈ 𝑀 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 Basecbs 16486 0gc0g 16716 1rcur 19254 Poly1cpl1 20348 coe1cco1 20349 deg1 cdg1 24651 Monic1pcmn1 24722 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-slot 16490 df-base 16492 df-mon1 24727 |
This theorem is referenced by: mon1puc1p 24747 deg1submon1p 24749 mon1psubm 39812 |
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