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Theorem facth1 26207

Description: The factor theorem and its converse. A polynomial 𝐹 has a root at 𝐴 iff 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹. (Contributed by Mario Carneiro, 12-Jun-2015.)

**Hypotheses**
Ref	Expression
ply1rem.p	⊢ 𝑃 = (Poly₁‘𝑅)
ply1rem.b	⊢ 𝐵 = (Base‘𝑃)
ply1rem.k	⊢ 𝐾 = (Base‘𝑅)
ply1rem.x	⊢ 𝑋 = (var₁‘𝑅)
ply1rem.m	⊢ − = (-_g‘𝑃)
ply1rem.a	⊢ 𝐴 = (algSc‘𝑃)
ply1rem.g	⊢ 𝐺 = (𝑋 − (𝐴‘𝑁))
ply1rem.o	⊢ 𝑂 = (eval₁‘𝑅)
ply1rem.1	⊢ (𝜑 → 𝑅 ∈ NzRing)
ply1rem.2	⊢ (𝜑 → 𝑅 ∈ CRing)
ply1rem.3	⊢ (𝜑 → 𝑁 ∈ 𝐾)
ply1rem.4	⊢ (𝜑 → 𝐹 ∈ 𝐵)
facth1.z	⊢ 0 = (0_g‘𝑅)
facth1.d	⊢ ∥ = (∥_r‘𝑃)

**Assertion**
Ref	Expression
facth1	⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))

**Proof of Theorem facth1**
Step	Hyp	Ref	Expression
1		ply1rem.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ NzRing)
2		nzrring 20517	. . . 4 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		ply1rem.4	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
5		ply1rem.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
6		ply1rem.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7		ply1rem.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
8		ply1rem.x	. . . . . 6 ⊢ 𝑋 = (var₁‘𝑅)
9		ply1rem.m	. . . . . 6 ⊢ − = (-_g‘𝑃)
10		ply1rem.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑃)
11		ply1rem.g	. . . . . 6 ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁))
12		ply1rem.o	. . . . . 6 ⊢ 𝑂 = (eval₁‘𝑅)
13		ply1rem.2	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
14		ply1rem.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐾)
15		eqid 2736	. . . . . 6 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
16		eqid 2736	. . . . . 6 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
17		facth1.z	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
18	5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17	ply1remlem 26205	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic_1p‘𝑅) ∧ ((deg₁‘𝑅)‘𝐺) = 1 ∧ (^◡(𝑂‘𝐺) “ { 0 }) = {𝑁}))
19	18	simp1d 1142	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic_1p‘𝑅))
20		eqid 2736	. . . . 5 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
21	20, 15	mon1puc1p 26191	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic_1p‘𝑅)) → 𝐺 ∈ (Unic_1p‘𝑅))
22	3, 19, 21	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Unic_1p‘𝑅))
23		facth1.d	. . . 4 ⊢ ∥ = (∥_r‘𝑃)
24		eqid 2736	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
25		eqid 2736	. . . 4 ⊢ (rem_1p‘𝑅) = (rem_1p‘𝑅)
26	5, 23, 6, 20, 24, 25	dvdsr1p 26204	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic_1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem_1p‘𝑅)𝐺) = (0_g‘𝑃)))
27	3, 4, 22, 26	syl3anc 1372	. 2 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ (𝐹(rem_1p‘𝑅)𝐺) = (0_g‘𝑃)))
28	5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 4, 25	ply1rem 26206	. . 3 ⊢ (𝜑 → (𝐹(rem_1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁)))
29	5, 10, 17, 24	ply1scl0 22294	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0_g‘𝑃))
30	3, 29	syl 17	. . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0_g‘𝑃))
31	30	eqcomd 2742	. . 3 ⊢ (𝜑 → (0_g‘𝑃) = (𝐴‘ 0 ))
32	28, 31	eqeq12d 2752	. 2 ⊢ (𝜑 → ((𝐹(rem_1p‘𝑅)𝐺) = (0_g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 )))
33	5, 10, 7, 6	ply1sclf1 22293	. . . 4 ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵)
34	3, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐴:𝐾–1-1→𝐵)
35	12, 5, 7, 6, 13, 14, 4	fveval1fvcl 22338	. . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾)
36	7, 17	ring0cl 20265	. . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
37	3, 36	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐾)
38		f1fveq 7283	. . 3 ⊢ ((𝐴:𝐾–1-1→𝐵 ∧ (((𝑂‘𝐹)‘𝑁) ∈ 𝐾 ∧ 0 ∈ 𝐾)) → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))
39	34, 35, 37, 38	syl12anc 836	. 2 ⊢ (𝜑 → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))
40	27, 32, 39	3bitrd 305	1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {csn 4625 class class class wbr 5142 ^◡ccnv 5683 “ cima 5687 –1-1→wf1 6557 ‘cfv 6560 (class class class)co 7432 1c1 11157 Basecbs 17248 0_gc0g 17485 -_gcsg 18954 Ringcrg 20231 CRingccrg 20232 ∥_rcdsr 20355 NzRingcnzr 20513 algSccascl 21873 var₁cv1 22178 Poly₁cpl1 22179 eval₁ce1 22319 deg₁cdg1 26094 Monic_1pcmn1 26166 Unic_1pcuc1p 26167 rem_1pcr1p 26169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-rhm 20473 df-nzr 20514 df-subrng 20547 df-subrg 20571 df-rlreg 20695 df-lmod 20861 df-lss 20931 df-lsp 20971 df-cnfld 21366 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-evls 22099 df-evl 22100 df-psr1 22182 df-vr1 22183 df-ply1 22184 df-coe1 22185 df-evl1 22321 df-mdeg 26095 df-deg1 26096 df-mon1 26171 df-uc1p 26172 df-q1p 26173 df-r1p 26174

This theorem is referenced by: fta1glem1 26208 fta1glem2 26209

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