Theorem evls1rhm 22209

Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)

Hypotheses

Ref	Expression
evls1rhm.q	⊢ 𝑄 = (𝑆 evalSub1 𝑅)
evls1rhm.b	⊢ 𝐵 = (Base‘𝑆)
evls1rhm.t	⊢ 𝑇 = (𝑆 ↑s 𝐵)
evls1rhm.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
evls1rhm.w	⊢ 𝑊 = (Poly1‘𝑈)

Assertion

Ref	Expression
evls1rhm	⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Proof of Theorem evls1rhm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1rhm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
2	1	subrgss 20481	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
3	2	adantl 481	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵)
4		elpwg 4566	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
5	4	adantl 481	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
6	3, 5	mpbird 257	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵)
7		evls1rhm.q	. . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
8		eqid 2729	. . . 4 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆)
9	7, 8, 1	evls1fval 22206	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
10	6, 9	syldan 591	. 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
11		evls1rhm.t	. . . 4 ⊢ 𝑇 = (𝑆 ↑s 𝐵)
12		eqid 2729	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
13	1, 11, 12	evls1rhmlem 22208	. . 3 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑m 1o)) RingHom 𝑇))
14		1on 8446	. . . . 5 ⊢ 1o ∈ On
15		eqid 2729	. . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
16		eqid 2729	. . . . . 6 ⊢ (1o mPoly 𝑈) = (1o mPoly 𝑈)
17		evls1rhm.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
18		eqid 2729	. . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 1o)) = (𝑆 ↑s (𝐵 ↑m 1o))
19	15, 16, 17, 18, 1	evlsrhm 21995	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o))))
20	14, 19	mp3an1 1450	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o))))
21		eqidd 2730	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊))
22		eqidd 2730	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑆 ↑s (𝐵 ↑m 1o))))
23		evls1rhm.w	. . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈)
24		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
25	23, 24	ply1bas 22079	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘(1o mPoly 𝑈))
26	25	a1i 11	. . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1o mPoly 𝑈)))
27		eqid 2729	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
28	23, 16, 27	ply1plusg 22108	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘(1o mPoly 𝑈))
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g‘𝑊) = (+g‘(1o mPoly 𝑈)))
30	29	oveqdr 7415	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1o mPoly 𝑈))𝑦))
31		eqidd 2730	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 1o))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 1o))))) → (𝑥(+g‘(𝑆 ↑s (𝐵 ↑m 1o)))𝑦) = (𝑥(+g‘(𝑆 ↑s (𝐵 ↑m 1o)))𝑦))
32		eqid 2729	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
33	23, 16, 32	ply1mulr 22110	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘(1o mPoly 𝑈))
34	33	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r‘𝑊) = (.r‘(1o mPoly 𝑈)))
35	34	oveqdr 7415	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1o mPoly 𝑈))𝑦))
36		eqidd 2730	. . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 1o))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 1o))))) → (𝑥(.r‘(𝑆 ↑s (𝐵 ↑m 1o)))𝑦) = (𝑥(.r‘(𝑆 ↑s (𝐵 ↑m 1o)))𝑦))
37	21, 22, 26, 22, 30, 31, 35, 36	rhmpropd 20518	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o))))
38	20, 37	eleqtrrd 2831	. . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 1o))))
39		rhmco 20410	. . 3 ⊢ (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑m 1o)) RingHom 𝑇) ∧ ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 1o)))) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
40	13, 38, 39	syl2an2r 685	. 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇))
41	10, 40	eqeltrd 2828	1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589   ↦ cmpt 5188   × cxp 5636   ∘ ccom 5642  Oncon0 6332  ‘cfv 6511  (class class class)co 7387  1_oc1o 8427   ↑_m cmap 8799  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221   ↑_s cpws 17409  CRingccrg 20143   RingHom crh 20378  SubRingcsubrg 20478   mPoly cmpl 21815   evalSub ces 21979  Poly₁cpl1 22061   evalSub₁ ces1 22200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  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 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-rhm 20381  df-subrng 20455  df-subrg 20479  df-lmod 20768  df-lss 20838  df-lsp 20878  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-psr1 22064  df-ply1 22066  df-evls1 22202

This theorem is referenced by:  evls1gsumadd  22211  evls1gsummul  22212  evls1pw  22213  evls1expd  22254  evls1fpws  22256  ressply1evl  22257  evls1fn  33529  evls1dm  33530  evls1fvf  33531  elirng  33681  irngnzply1lem  33685  irngnzply1  33686