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Theorem evlsscasrng 22060

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 17270	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾_s 𝐵) = 𝑆)
6	5	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾_s 𝐵))
7	3, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾_s 𝐵))
8	7	oveq2d 7426	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑆) = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
9	2, 8	eqtrid 2783	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
10	9	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
11	1, 10	eqtrid 2783	. . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
12	11	fveq1d 6883	. . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋))
13	12	fveq2d 6885	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)))
14		eqid 2736	. . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵)
15		eqid 2736	. . . 4 ⊢ (𝐼 mPoly (𝑆 ↾_s 𝐵)) = (𝐼 mPoly (𝑆 ↾_s 𝐵))
16		eqid 2736	. . . 4 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
17		eqid 2736	. . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))
18		evlsscasrng.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
19		crngring 20210	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
20	4	subrgid 20538	. . . . 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 20537	. . . . . 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 3964	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
27	14, 15, 16, 4, 17, 18, 3, 21, 26	evlssca 22052	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
28	13, 27	eqtrd 2771	. 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
29		evlsscasrng.o	. . . . 5 ⊢ 𝑂 = (𝐼 eval 𝑆)
30	29, 4	evlval 22058	. . . 4 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)
31	30	a1i 11	. . 3 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵))
32	31	fveq1d 6883	. 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 22052	. 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
38	28, 32, 37	3eqtr4rd 2782	1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⊆ wss 3931 {csn 4606 × cxp 5657 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 Basecbs 17233 ↾_s cress 17256 Ringcrg 20198 CRingccrg 20199 SubRingcsubrg 20534 algSccascl 21817 mPoly cmpl 21871 evalSub ces 22035 eval cevl 22036

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-srg 20152 df-ring 20200 df-cring 20201 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-lmod 20824 df-lss 20894 df-lsp 20934 df-assa 21818 df-asp 21819 df-ascl 21820 df-psr 21874 df-mvr 21875 df-mpl 21876 df-evls 22037 df-evl 22038

This theorem is referenced by: evlsca 22061

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