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Theorem facth1 24849

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 20087	. . . 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 2759	. . . . . 6 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
16		eqid 2759	. . . . . 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 24847	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic_1p‘𝑅) ∧ (( deg₁ ‘𝑅)‘𝐺) = 1 ∧ (^◡(𝑂‘𝐺) “ { 0 }) = {𝑁}))
19	18	simp1d 1140	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic_1p‘𝑅))
20		eqid 2759	. . . . 5 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
21	20, 15	mon1puc1p 24835	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic_1p‘𝑅)) → 𝐺 ∈ (Unic_1p‘𝑅))
22	3, 19, 21	syl2anc 588	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Unic_1p‘𝑅))
23		facth1.d	. . . 4 ⊢ ∥ = (∥_r‘𝑃)
24		eqid 2759	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
25		eqid 2759	. . . 4 ⊢ (rem_1p‘𝑅) = (rem_1p‘𝑅)
26	5, 23, 6, 20, 24, 25	dvdsr1p 24846	. . 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 1369	. 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 24848	. . 3 ⊢ (𝜑 → (𝐹(rem_1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁)))
29	5, 10, 17, 24	ply1scl0 20999	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0_g‘𝑃))
30	3, 29	syl 17	. . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0_g‘𝑃))
31	30	eqcomd 2765	. . 3 ⊢ (𝜑 → (0_g‘𝑃) = (𝐴‘ 0 ))
32	28, 31	eqeq12d 2775	. 2 ⊢ (𝜑 → ((𝐹(rem_1p‘𝑅)𝐺) = (0_g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 )))
33	5, 10, 7, 6	ply1sclf1 20998	. . . 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 21037	. . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾)
36	7, 17	ring0cl 19375	. . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
37	3, 36	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐾)
38		f1fveq 7005	. . 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 309	1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 = wceq 1539 ∈ wcel 2112 {csn 4515 class class class wbr 5025 ^◡ccnv 5516 “ cima 5520 –1-1→wf1 6325 ‘cfv 6328 (class class class)co 7143 1c1 10561 Basecbs 16526 0_gc0g 16756 -_gcsg 18156 Ringcrg 19350 CRingccrg 19351 ∥_rcdsr 19444 NzRingcnzr 20083 algSccascl 20602 var₁cv1 20885 Poly₁cpl1 20886 eval₁ce1 21018 deg₁ cdg1 24736 Monic_1pcmn1 24810 Unic_1pcuc1p 24811 rem_1pcr1p 24813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 ax-addf 10639 ax-mulf 10640

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-ofr 7399 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-sup 8924 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-fzo 13068 df-seq 13404 df-hash 13726 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-hom 16632 df-cco 16633 df-0g 16758 df-gsum 16759 df-prds 16764 df-pws 16766 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-mhm 18007 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-mulg 18277 df-subg 18328 df-ghm 18408 df-cntz 18499 df-cmn 18960 df-abl 18961 df-mgp 19293 df-ur 19305 df-srg 19309 df-ring 19352 df-cring 19353 df-oppr 19429 df-dvdsr 19447 df-unit 19448 df-invr 19478 df-rnghom 19523 df-subrg 19586 df-lmod 19689 df-lss 19757 df-lsp 19797 df-nzr 20084 df-rlreg 20109 df-cnfld 20152 df-assa 20603 df-asp 20604 df-ascl 20605 df-psr 20656 df-mvr 20657 df-mpl 20658 df-opsr 20660 df-evls 20820 df-evl 20821 df-psr1 20889 df-vr1 20890 df-ply1 20891 df-coe1 20892 df-evl1 21020 df-mdeg 24737 df-deg1 24738 df-mon1 24815 df-uc1p 24816 df-q1p 24817 df-r1p 24818

This theorem is referenced by: fta1glem1 24850 fta1glem2 24851

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