uc1pldg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uc1pldg		Structured version Visualization version GIF version

Theorem uc1pldg 25901

Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)

**Hypotheses**
Ref	Expression
uc1pldg.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
uc1pldg.u	⊢ 𝑈 = (Unit‘𝑅)
uc1pldg.c	⊢ 𝐶 = (Unic_1p‘𝑅)

**Assertion**
Ref	Expression
uc1pldg	⊢ (𝐹 ∈ 𝐶 → ((coe₁‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

**Proof of Theorem uc1pldg**
Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
2		eqid 2730	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
3		eqid 2730	. . 3 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
4		uc1pldg.d	. . 3 ⊢ 𝐷 = ( deg₁ ‘𝑅)
5		uc1pldg.c	. . 3 ⊢ 𝐶 = (Unic_1p‘𝑅)
6		uc1pldg.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
7	1, 2, 3, 4, 5, 6	isuc1p 25893	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly₁‘𝑅)) ∧ 𝐹 ≠ (0_g‘(Poly₁‘𝑅)) ∧ ((coe₁‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))
8	7	simp3bi 1145	1 ⊢ (𝐹 ∈ 𝐶 → ((coe₁‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ‘cfv 6542 Basecbs 17148 0_gc0g 17389 Unitcui 20246 Poly₁cpl1 21920 coe₁cco1 21921 deg₁ cdg1 25804 Unic_1pcuc1p 25879

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-nn 12217 df-slot 17119 df-ndx 17131 df-base 17149 df-uc1p 25884

This theorem is referenced by: uc1pmon1p 25904 q1peqb 25907 fta1glem1 25918 ig1peu 25924

W3C validator