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Theorem evlsscasrng 22021

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 17210	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾_s 𝐵) = 𝑆)
6	5	eqcomd 2733	. . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾_s 𝐵))
7	3, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾_s 𝐵))
8	7	oveq2d 7430	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑆) = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
9	2, 8	eqtrid 2779	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐼 mPoly (𝑆 ↾_s 𝐵)))
10	9	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
11	1, 10	eqtrid 2779	. . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))))
12	11	fveq1d 6893	. . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋))
13	12	fveq2d 6895	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)))
14		eqid 2727	. . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵)
15		eqid 2727	. . . 4 ⊢ (𝐼 mPoly (𝑆 ↾_s 𝐵)) = (𝐼 mPoly (𝑆 ↾_s 𝐵))
16		eqid 2727	. . . 4 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
17		eqid 2727	. . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵))) = (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))
18		evlsscasrng.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
19		crngring 20169	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
20	4	subrgid 20494	. . . . 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 20493	. . . . . 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 3979	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
27	14, 15, 16, 4, 17, 18, 3, 21, 26	evlssca 22013	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝐵)))‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
28	13, 27	eqtrd 2767	. 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
29		evlsscasrng.o	. . . . 5 ⊢ 𝑂 = (𝐼 eval 𝑆)
30	29, 4	evlval 22019	. . . 4 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)
31	30	a1i 11	. . 3 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵))
32	31	fveq1d 6893	. 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 22013	. 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑_m 𝐼) × {𝑋}))
38	28, 32, 37	3eqtr4rd 2778	1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 {csn 4624 × cxp 5670 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8834 Basecbs 17165 ↾_s cress 17194 Ringcrg 20157 CRingccrg 20158 SubRingcsubrg 20488 algSccascl 21766 mPoly cmpl 21819 evalSub ces 21994 eval cevl 21995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-lsp 20838 df-assa 21767 df-asp 21768 df-ascl 21769 df-psr 21822 df-mvr 21823 df-mpl 21824 df-evls 21996 df-evl 21997

This theorem is referenced by: evlsca 22022

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