evls1subd - Mathbox for Thierry Arnoux

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Theorem evls1subd 33729

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 7404	. . . . . 6 ⊢ (𝑀𝐷𝑁) = (𝑀(-_g‘𝑊)𝑁)
3		eqid 2761	. . . . . . 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 2761	. . . . . . 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 33727	. . . . . 6 ⊢ (𝜑 → (𝑀(-_g‘𝑊)𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
12	2, 11	eqtrid 2808	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
13	3, 4, 5, 6	subrgply1 22282	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubRing‘(Poly₁‘𝑆)))
14		subrgsubg 20614	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(Poly₁‘𝑆)) → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
15	7, 13, 14	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘(Poly₁‘𝑆)))
16		eqid 2761	. . . . . . 7 ⊢ (-_g‘(Poly₁‘𝑆)) = (-_g‘(Poly₁‘𝑆))
17		eqid 2761	. . . . . . 7 ⊢ (-_g‘((Poly₁‘𝑆) ↾_s 𝐵)) = (-_g‘((Poly₁‘𝑆) ↾_s 𝐵))
18	16, 8, 17	subgsub 19171	. . . . . 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 1389	. . . . 5 ⊢ (𝜑 → (𝑀(-_g‘(Poly₁‘𝑆))𝑁) = (𝑀(-_g‘((Poly₁‘𝑆) ↾_s 𝐵))𝑁))
20	12, 19	eqtr4d 2799	. . . 4 ⊢ (𝜑 → (𝑀𝐷𝑁) = (𝑀(-_g‘(Poly₁‘𝑆))𝑁))
21	20	fveq2d 6866	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁)))
22	21	fveq1d 6864	. 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 2761	. . . . . 6 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
26		evls1subd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
27	23, 24, 5, 4, 6, 25, 26, 7	ressply1evl 22421	. . . . 5 ⊢ (𝜑 → 𝑄 = ((eval₁‘𝑆) ↾ 𝐵))
28	27	fveq1d 6864	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑀𝐷𝑁)) = (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)))
29	4	subrgring 20611	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
30	5	ply1ring 22297	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
31	7, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
32	31	ringgrpd 20279	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Grp)
33	6, 1	grpsubcl 19053	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀𝐷𝑁) ∈ 𝐵)
34	32, 9, 10, 33	syl3anc 1389	. . . . 5 ⊢ (𝜑 → (𝑀𝐷𝑁) ∈ 𝐵)
35	34	fvresd 6882	. . . 4 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘(𝑀𝐷𝑁)) = ((eval₁‘𝑆)‘(𝑀𝐷𝑁)))
36	28, 35	eqtr2d 2797	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(𝑀𝐷𝑁)) = (𝑄‘(𝑀𝐷𝑁)))
37	36	fveq1d 6864	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀𝐷𝑁))‘𝐶) = ((𝑄‘(𝑀𝐷𝑁))‘𝐶))
38		eqid 2761	. . . 4 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
39		evls1subd.y	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
40		eqid 2761	. . . . . . . 8 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
41		eqid 2761	. . . . . . . 8 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
42	3, 4, 5, 6, 7, 40, 41, 38	ressply1bas2 22277	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
43		inss2 4187	. . . . . . 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 3935	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly₁‘𝑆)))
46	27	fveq1d 6864	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑀))
47	9	fvresd 6882	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑀) = ((eval₁‘𝑆)‘𝑀))
48	46, 47	eqtr2d 2797	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑀) = (𝑄‘𝑀))
49	48	fveq1d 6864	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))
50	45, 49	jca 519	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly₁‘𝑆)) ∧ (((eval₁‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)))
51	44, 10	sseldd 3935	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly₁‘𝑆)))
52	27	fveq1d 6864	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval₁‘𝑆) ↾ 𝐵)‘𝑁))
53	10	fvresd 6882	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝑆) ↾ 𝐵)‘𝑁) = ((eval₁‘𝑆)‘𝑁))
54	52, 53	eqtr2d 2797	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝑆)‘𝑁) = (𝑄‘𝑁))
55	54	fveq1d 6864	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))
56	51, 55	jca 519	. . . 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 22393	. . 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 499	. 2 ⊢ (𝜑 → (((eval₁‘𝑆)‘(𝑀(-_g‘(Poly₁‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))
60	22, 37, 59	3eqtr3d 2804	1 ⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ∩ cin 3901 ↾ cres 5645 ‘cfv 6516 (class class class)co 7391 Basecbs 17236 ↾_s cress 17257 Grpcgrp 18966 -_gcsg 18968 SubGrpcsubg 19153 Ringcrg 20270 CRingccrg 20271 SubRingcsubrg 20606 PwSer₁cps1 22225 Poly₁cpl1 22227 evalSub₁ ces1 22364 eval₁ce1 22365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-ofr 7656 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-sup 9382 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-fz 13507 df-fzo 13654 df-seq 14009 df-hash 14338 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-hom 17301 df-cco 17302 df-0g 17461 df-gsum 17462 df-prds 17467 df-pws 17469 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-subg 19156 df-ghm 19245 df-cntz 19348 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-srg 20224 df-ring 20272 df-cring 20273 df-rhm 20508 df-subrng 20583 df-subrg 20607 df-lmod 20917 df-lss 20987 df-lsp 21027 df-assa 21893 df-asp 21894 df-ascl 21895 df-psr 21949 df-mvr 21950 df-mpl 21951 df-opsr 21953 df-evls 22115 df-evl 22116 df-psr1 22230 df-vr1 22231 df-ply1 22232 df-coe1 22233 df-evls1 22366 df-evl1 22367

This theorem is referenced by: irredminply 33974 2sqr3minply 34038

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