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Theorem evlsscasrng 19749

Description: The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 12-Sep-2019.)

**Hypotheses**
Ref	Expression
evlsscasrng.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsscasrng.o	⊢ 𝑂 = (𝐼 eval 𝑆)
evlsscasrng.w	⊢ 𝑊 = (𝐼 mPoly 𝑈)
evlsscasrng.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsscasrng.p	⊢ 𝑃 = (𝐼 mPoly 𝑆)
evlsscasrng.b	⊢ 𝐵 = (Base‘𝑆)
evlsscasrng.a	⊢ 𝐴 = (algSc‘𝑊)
evlsscasrng.c	⊢ 𝐶 = (algSc‘𝑃)
evlsscasrng.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsscasrng.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsscasrng.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsscasrng.x	⊢ (𝜑 → 𝑋 ∈ 𝑅)

**Assertion**
Ref	Expression
evlsscasrng	⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋)))

**Proof of Theorem evlsscasrng**
Step	Hyp	Ref	Expression
1		evlsscasrng.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑃)
2		evlsscasrng.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑆)
3		evlsscasrng.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ CRing)
4		evlsscasrng.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑆)
5	4	ressid 16158	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾_s 𝐵) = 𝑆)
6	5	eqcomd 2767	. . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾_s 𝐵))
7	3, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾_s 𝐵))
8	7	oveq2d 6831	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑆) = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
9	2, 8	syl5eq 2807	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
10	9	fveq2d 6358	. . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
11	1, 10	syl5eq 2807	. . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
12	11	fveq1d 6356	. . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋))
13	12	fveq2d 6358	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)))
14		eqid 2761	. . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵)
15		eqid 2761	. . . 4 ⊢ (𝐼 mPoly (𝑆 ↾_s 𝐵)) = (𝐼 mPoly (𝑆 ↾_s 𝐵))
16		eqid 2761	. . . 4 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
17		eqid 2761	. . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))
18		evlsscasrng.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
19		crngring 18779	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
20	4	subrgid 19005	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆))
21	3, 19, 20	3syl 18	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆))
22		evlsscasrng.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
23	4	subrgss 19004	. . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ 𝐵)
25		evlsscasrng.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑅)
26	24, 25	sseldd 3746	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
27	14, 15, 16, 4, 17, 18, 3, 21, 26	evlssca 19745	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)) = ((𝐵 ↑_𝑚 𝐼) × {𝑋}))
28	13, 27	eqtrd 2795	. 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = ((𝐵 ↑_𝑚 𝐼) × {𝑋}))
29		evlsscasrng.o	. . . . 5 ⊢ 𝑂 = (𝐼 eval 𝑆)
30	29, 4	evlval 19747	. . . 4 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)
31	30	a1i 11	. . 3 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵))
32	31	fveq1d 6356	. 2 ⊢ (𝜑 → (𝑂‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)))
33		evlsscasrng.q	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
34		evlsscasrng.w	. . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
35		evlsscasrng.u	. . 3 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
36		evlsscasrng.a	. . 3 ⊢ 𝐴 = (algSc‘𝑊)
37	33, 34, 35, 4, 36, 18, 3, 22, 25	evlssca 19745	. 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑_𝑚 𝐼) × {𝑋}))
38	28, 32, 37	3eqtr4rd 2806	1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2140 ⊆ wss 3716 {csn 4322 × cxp 5265 ‘cfv 6050 (class class class)co 6815 ↑_𝑚 cmap 8026 Basecbs 16080 ↾_s cress 16081 Ringcrg 18768 CRingccrg 18769 SubRingcsubrg 18999 algSccascl 19534 mPoly cmpl 19576 evalSub ces 19727 eval cevl 19728

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-of 7064 df-ofr 7065 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-2o 7732 df-oadd 7735 df-er 7914 df-map 8028 df-pm 8029 df-ixp 8078 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-sup 8516 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-fz 12541 df-fzo 12681 df-seq 13017 df-hash 13333 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-sca 16180 df-vsca 16181 df-ip 16182 df-tset 16183 df-ple 16184 df-ds 16187 df-hom 16189 df-cco 16190 df-0g 16325 df-gsum 16326 df-prds 16331 df-pws 16333 df-mre 16469 df-mrc 16470 df-acs 16472 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-mhm 17557 df-submnd 17558 df-grp 17647 df-minusg 17648 df-sbg 17649 df-mulg 17763 df-subg 17813 df-ghm 17880 df-cntz 17971 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-srg 18727 df-ring 18770 df-cring 18771 df-rnghom 18938 df-subrg 19001 df-lmod 19088 df-lss 19156 df-lsp 19195 df-assa 19535 df-asp 19536 df-ascl 19537 df-psr 19579 df-mvr 19580 df-mpl 19581 df-evls 19729 df-evl 19730

This theorem is referenced by: evlsca 19750

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