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Theorem facth1 26145

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 20466	. . . 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 26143	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic_1p‘𝑅) ∧ ((deg₁‘𝑅)‘𝐺) = 1 ∧ (^◡(𝑂‘𝐺) “ { 0 }) = {𝑁}))
19	18	simp1d 1143	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic_1p‘𝑅))
20		eqid 2737	. . . . 5 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
21	20, 15	mon1puc1p 26129	. . . 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 26142	. . 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 1374	. 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 26144	. . 3 ⊢ (𝜑 → (𝐹(rem_1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁)))
29	5, 10, 17, 24	ply1scl0 22249	. . . . 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 22248	. . . 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 22294	. . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾)
36	7, 17	ring0cl 20219	. . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
37	3, 36	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐾)
38		f1fveq 7220	. . 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 837	. 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 1542 ∈ wcel 2114 {csn 4582 class class class wbr 5100 ^◡ccnv 5633 “ cima 5637 –1-1→wf1 6499 ‘cfv 6502 (class class class)co 7370 1c1 11041 Basecbs 17150 0_gc0g 17373 -_gcsg 18882 Ringcrg 20185 CRingccrg 20186 ∥_rcdsr 20307 NzRingcnzr 20462 algSccascl 21824 var₁cv1 22133 Poly₁cpl1 22134 eval₁ce1 22275 deg₁cdg1 26032 Monic_1pcmn1 26104 Unic_1pcuc1p 26105 rem_1pcr1p 26107

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-ofr 7635 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-tpos 8180 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-sup 9359 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-fzo 13585 df-seq 13939 df-hash 14268 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-0g 17375 df-gsum 17376 df-prds 17381 df-pws 17383 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-submnd 18723 df-grp 18883 df-minusg 18884 df-sbg 18885 df-mulg 19015 df-subg 19070 df-ghm 19159 df-cntz 19263 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-srg 20139 df-ring 20187 df-cring 20188 df-oppr 20290 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-rhm 20425 df-nzr 20463 df-subrng 20496 df-subrg 20520 df-rlreg 20644 df-lmod 20830 df-lss 20900 df-lsp 20940 df-cnfld 21327 df-assa 21825 df-asp 21826 df-ascl 21827 df-psr 21882 df-mvr 21883 df-mpl 21884 df-opsr 21886 df-evls 22046 df-evl 22047 df-psr1 22137 df-vr1 22138 df-ply1 22139 df-coe1 22140 df-evl1 22277 df-mdeg 26033 df-deg1 26034 df-mon1 26109 df-uc1p 26110 df-q1p 26111 df-r1p 26112

This theorem is referenced by: fta1glem1 26146 fta1glem2 26147

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