evls1subd - Mathbox for Thierry Arnoux

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Theorem evls1subd 33632

Description: Univariate polynomial evaluation of a difference of polynomials. (Contributed by Thierry Arnoux, 25-Apr-2025.)

**Hypotheses**
Ref	Expression
ressply1evl.q	⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
ressply1evl.k	⊢ 𝐾 = (Base‘𝑆)
ressply1evl.w	⊢ 𝑊 = (Poly₁‘𝑈)
ressply1evl.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
ressply1evl.b	⊢ 𝐵 = (Base‘𝑊)
evls1subd.1	⊢ 𝐷 = (-_g‘𝑊)
evls1subd.2	⊢ − = (-_g‘𝑆)
evls1subd.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1subd.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1subd.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
evls1subd.n	⊢ (𝜑 → 𝑁 ∈ 𝐵)
evls1subd.y	⊢ (𝜑 → 𝐶 ∈ 𝐾)

**Assertion**
Ref	Expression
evls1subd	⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))

**Proof of Theorem evls1subd**
Step	Hyp	Ref	Expression
1		evls1subd.1	. . . . . . 7 ⊢ 𝐷 = (-_g‘𝑊)
2	1	oveqi 7380	. . . . . 6 ⊢ (𝑀𝐷𝑁) = (𝑀(-_g‘𝑊)𝑁)
3		eqid 2736	. . . . . . 7 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
4		ressply1evl.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
5		ressply1evl.w	. . . . . . 7 ⊢ 𝑊 = (Poly₁‘𝑈)
6		ressply1evl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
7		evls1subd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
8		eqid 2736	. . . . . . 7 ⊢ ((Poly₁‘𝑆) ↾_s 𝐵) = ((Poly₁‘𝑆) ↾_s 𝐵)
9		evls1subd.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
10		evls1subd.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
11	3, 4, 5, 6, 7, 8, 9, 10	ressply1sub 33630	. . . . . 6 ⊢ (𝜑 → (𝑀(-_g‘𝑊)𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12	2, 11	eqtrid 2783	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
13	3, 4, 5, 6	subrgply1 22196	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubRing‘(Poly₁‘𝑆)))
14		subrgsubg 20554	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(Poly₁‘𝑆)) → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
15	7, 13, 14	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
16		eqid 2736	. . . . . . 7 ⊢ (-_g‘(Poly₁‘𝑆)) = (-_g‘(Poly₁‘𝑆))
17		eqid 2736	. . . . . . 7 ⊢ (-_g‘((Poly₁‘𝑆) ↾_s 𝐵)) = (-_g‘((Poly₁‘𝑆) ↾_s 𝐵))
18	16, 8, 17	subgsub 19114	. . . . . 6 ⊢ ((𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀(-_g‘(Poly₁‘𝑆))𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
19	15, 9, 10, 18	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑀(-_g‘(Poly₁‘𝑆))𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
20	12, 19	eqtr4d 2774	. . . 4 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘(Poly₁‘𝑆))𝑁))
21	20	fveq2d 6844	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁)))
22	21	fveq1d 6842	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀𝐷𝑁))‘𝐶) = (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶))
23		ressply1evl.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
24		ressply1evl.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
25		eqid 2736	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
26		evls1subd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
27	23, 24, 5, 4, 6, 25, 26, 7	ressply1evl 22335	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
28	27	fveq1d 6842	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀𝐷𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)))
29	4	subrgring 20551	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
30	5	ply1ring 22211	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
31	7, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
32	31	ringgrpd 20223	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
33	6, 1	grpsubcl 18996	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀𝐷𝑁) ∈ 𝐵)
34	32, 9, 10, 33	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) ∈ 𝐵)
35	34	fvresd 6860	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀𝐷𝑁)))
36	28, 35	eqtr2d 2772	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = (𝑄‘(𝑀𝐷𝑁)))
37	36	fveq1d 6842	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀𝐷𝑁))‘𝐶) = ((𝑄‘(𝑀𝐷𝑁))‘𝐶))
38		eqid 2736	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
39		evls1subd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
40		eqid 2736	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
41		eqid 2736	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
42	3, 4, 5, 6, 7, 40, 41, 38	ressply1bas2 22191	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
43		inss2 4178	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
44	42, 43	eqsstrdi 3966	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
45	44, 9	sseldd 3922	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
46	27	fveq1d 6842	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
47	9	fvresd 6860	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
48	46, 47	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
49	48	fveq1d 6842	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
50	45, 49	jca 511	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
51	44, 10	sseldd 3922	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
52	27	fveq1d 6842	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
53	10	fvresd 6860	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
54	52, 53	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
55	54	fveq1d 6842	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
56	51, 55	jca 511	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)))
57		evls1subd.2	. . . 4 ⊢ − = (-_g‘𝑆)
58	25, 3, 24, 38, 26, 39, 50, 56, 16, 57	evl1subd 22307	. . 3 ⊢ (𝜑 → ((𝑀(-_g‘(Poly₁‘𝑆))𝑁) ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶))))
59	58	simprd 495	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))
60	22, 37, 59	3eqtr3d 2779	1 ⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∩ cin 3888 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 ↾_s cress 17200 Grpcgrp 18909 -_gcsg 18911 SubGrpcsubg 19096 Ringcrg 20214 CRingccrg 20215 SubRingcsubrg 20546 PwSer₁cps1 22138 Poly₁cpl1 22140 evalSub₁ ces1 22278 eval₁ce1 22279

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-srg 20168 df-ring 20216 df-cring 20217 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-lmod 20857 df-lss 20927 df-lsp 20967 df-assa 21833 df-asp 21834 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-evls 22052 df-evl 22053 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-evls1 22280 df-evl1 22281

This theorem is referenced by: irredminply 33860 2sqr3minply 33924

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