evls1subd - Mathbox for Thierry Arnoux

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Theorem evls1subd 33653

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 7371	. . . . . 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 33651	. . . . . 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 22173	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubRing‘(Poly₁‘𝑆)))
14		subrgsubg 20510	. . . . . . 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 19068	. . . . . 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 1373	. . . . 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 6838	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁)))
22	21	fveq1d 6836	. 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 22314	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
28	27	fveq1d 6836	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀𝐷𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)))
29	4	subrgring 20507	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
30	5	ply1ring 22188	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
31	7, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
32	31	ringgrpd 20177	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
33	6, 1	grpsubcl 18950	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀𝐷𝑁) ∈ 𝐵)
34	32, 9, 10, 33	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) ∈ 𝐵)
35	34	fvresd 6854	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀𝐷𝑁)))
36	28, 35	eqtr2d 2772	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = (𝑄‘(𝑀𝐷𝑁)))
37	36	fveq1d 6836	. 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 22168	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
43		inss2 4190	. . . . . . 7 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
44	42, 43	eqsstrdi 3978	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
45	44, 9	sseldd 3934	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
46	27	fveq1d 6836	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
47	9	fvresd 6854	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
48	46, 47	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
49	48	fveq1d 6836	. . . . 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 3934	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
52	27	fveq1d 6836	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
53	10	fvresd 6854	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
54	52, 53	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
55	54	fveq1d 6836	. . . . 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 22286	. . 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 1541 ∈ wcel 2113 ∩ cin 3900 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 ↾_s cress 17157 Grpcgrp 18863 -_gcsg 18865 SubGrpcsubg 19050 Ringcrg 20168 CRingccrg 20169 SubRingcsubrg 20502 PwSer₁cps1 22115 Poly₁cpl1 22117 evalSub₁ ces1 22257 eval₁ce1 22258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-srg 20122 df-ring 20170 df-cring 20171 df-rhm 20408 df-subrng 20479 df-subrg 20503 df-lmod 20813 df-lss 20883 df-lsp 20923 df-assa 21808 df-asp 21809 df-ascl 21810 df-psr 21865 df-mvr 21866 df-mpl 21867 df-opsr 21869 df-evls 22029 df-evl 22030 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-evls1 22259 df-evl1 22260

This theorem is referenced by: irredminply 33873 2sqr3minply 33937

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