Theorem ismon1p 24122

Description: Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

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)

Assertion

Ref	Expression
ismon1p	⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))

Proof of Theorem ismon1p

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 3005	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 ))
2		fveq2 6332	. . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹))
3		fveq2 6332	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
4	2, 3	fveq12d 6338	. . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹)))
5	4	eqeq1d 2773	. . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))
6	1, 5	anbi12d 616	. . 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 24121	. . 3 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}
14	6, 13	elrab2 3518	. 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )))
15		3anass 1080	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )))
16	14, 15	bitr4i 267	1 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ‘cfv 6031  Basecbs 16064  0_gc0g 16308  1_rcur 18709  Poly₁cpl1 19762  coe₁cco1 19763   deg₁ cdg1 24034  Monic_1pcmn1 24105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16068  df-base 16070  df-mon1 24110

This theorem is referenced by:  mon1pcl  24124  mon1pn0  24126  mon1pldg  24129  uc1pmon1p  24131  ply1remlem  24142  mon1pid  38309  mon1psubm  38310