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Theorem facth1 25435

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 20638	. . . 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 2737	. . . . . 6 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
16		eqid 2737	. . . . . 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 25433	. . . . 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 2737	. . . . 5 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
21	20, 15	mon1puc1p 25421	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic_1p‘𝑅)) → 𝐺 ∈ (Unic_1p‘𝑅))
22	3, 19, 21	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Unic_1p‘𝑅))
23		facth1.d	. . . 4 ⊢ ∥ = (∥_r‘𝑃)
24		eqid 2737	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
25		eqid 2737	. . . 4 ⊢ (rem_1p‘𝑅) = (rem_1p‘𝑅)
26	5, 23, 6, 20, 24, 25	dvdsr1p 25432	. . 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 1371	. 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 25434	. . 3 ⊢ (𝜑 → (𝐹(rem_1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁)))
29	5, 10, 17, 24	ply1scl0 21567	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0_g‘𝑃))
30	3, 29	syl 17	. . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0_g‘𝑃))
31	30	eqcomd 2743	. . 3 ⊢ (𝜑 → (0_g‘𝑃) = (𝐴‘ 0 ))
32	28, 31	eqeq12d 2753	. 2 ⊢ (𝜑 → ((𝐹(rem_1p‘𝑅)𝐺) = (0_g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 )))
33	5, 10, 7, 6	ply1sclf1 21566	. . . 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 21605	. . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾)
36	7, 17	ring0cl 19903	. . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
37	3, 36	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐾)
38		f1fveq 7196	. . 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 835	. 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 205 = wceq 1541 ∈ wcel 2106 {csn 4578 class class class wbr 5097 ^◡ccnv 5624 “ cima 5628 –1-1→wf1 6481 ‘cfv 6484 (class class class)co 7342 1c1 10978 Basecbs 17010 0_gc0g 17248 -_gcsg 18676 Ringcrg 19878 CRingccrg 19879 ∥_rcdsr 19975 NzRingcnzr 20634 algSccascl 21165 var₁cv1 21453 Poly₁cpl1 21454 eval₁ce1 21586 deg₁ cdg1 25322 Monic_1pcmn1 25396 Unic_1pcuc1p 25397 rem_1pcr1p 25399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5234 ax-sep 5248 ax-nul 5255 ax-pow 5313 ax-pr 5377 ax-un 7655 ax-cnex 11033 ax-resscn 11034 ax-1cn 11035 ax-icn 11036 ax-addcl 11037 ax-addrcl 11038 ax-mulcl 11039 ax-mulrcl 11040 ax-mulcom 11041 ax-addass 11042 ax-mulass 11043 ax-distr 11044 ax-i2m1 11045 ax-1ne0 11046 ax-1rid 11047 ax-rnegex 11048 ax-rrecex 11049 ax-cnre 11050 ax-pre-lttri 11051 ax-pre-lttrn 11052 ax-pre-ltadd 11053 ax-pre-mulgt0 11054 ax-pre-sup 11055 ax-addf 11056 ax-mulf 11057

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3732 df-csb 3848 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3921 df-nul 4275 df-if 4479 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4858 df-int 4900 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5181 df-tr 5215 df-id 5523 df-eprel 5529 df-po 5537 df-so 5538 df-fr 5580 df-se 5581 df-we 5582 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6243 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6436 df-fun 6486 df-fn 6487 df-f 6488 df-f1 6489 df-fo 6490 df-f1o 6491 df-fv 6492 df-isom 6493 df-riota 7298 df-ov 7345 df-oprab 7346 df-mpo 7347 df-of 7600 df-ofr 7601 df-om 7786 df-1st 7904 df-2nd 7905 df-supp 8053 df-tpos 8117 df-frecs 8172 df-wrecs 8203 df-recs 8277 df-rdg 8316 df-1o 8372 df-er 8574 df-map 8693 df-pm 8694 df-ixp 8762 df-en 8810 df-dom 8811 df-sdom 8812 df-fin 8813 df-fsupp 9232 df-sup 9304 df-oi 9372 df-card 9801 df-pnf 11117 df-mnf 11118 df-xr 11119 df-ltxr 11120 df-le 11121 df-sub 11313 df-neg 11314 df-nn 12080 df-2 12142 df-3 12143 df-4 12144 df-5 12145 df-6 12146 df-7 12147 df-8 12148 df-9 12149 df-n0 12340 df-z 12426 df-dec 12544 df-uz 12689 df-fz 13346 df-fzo 13489 df-seq 13828 df-hash 14151 df-struct 16946 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-mulr 17074 df-starv 17075 df-sca 17076 df-vsca 17077 df-ip 17078 df-tset 17079 df-ple 17080 df-ds 17082 df-unif 17083 df-hom 17084 df-cco 17085 df-0g 17250 df-gsum 17251 df-prds 17256 df-pws 17258 df-mre 17393 df-mrc 17394 df-acs 17396 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-mhm 18528 df-submnd 18529 df-grp 18677 df-minusg 18678 df-sbg 18679 df-mulg 18798 df-subg 18849 df-ghm 18929 df-cntz 19020 df-cmn 19484 df-abl 19485 df-mgp 19816 df-ur 19833 df-srg 19837 df-ring 19880 df-cring 19881 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-rnghom 20054 df-subrg 20127 df-lmod 20231 df-lss 20300 df-lsp 20340 df-nzr 20635 df-rlreg 20660 df-cnfld 20704 df-assa 21166 df-asp 21167 df-ascl 21168 df-psr 21218 df-mvr 21219 df-mpl 21220 df-opsr 21222 df-evls 21388 df-evl 21389 df-psr1 21457 df-vr1 21458 df-ply1 21459 df-coe1 21460 df-evl1 21588 df-mdeg 25323 df-deg1 25324 df-mon1 25401 df-uc1p 25402 df-q1p 25403 df-r1p 25404

This theorem is referenced by: fta1glem1 25436 fta1glem2 25437

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