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Theorem evl1sca 22249

Description: Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)

**Hypotheses**
Ref	Expression
evl1sca.o	⊢ 𝑂 = (eval₁‘𝑅)
evl1sca.p	⊢ 𝑃 = (Poly₁‘𝑅)
evl1sca.b	⊢ 𝐵 = (Base‘𝑅)
evl1sca.a	⊢ 𝐴 = (algSc‘𝑃)

**Assertion**
Ref	Expression
evl1sca	⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Proof of Theorem evl1sca Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20163	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring)
3		evl1sca.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
4		evl1sca.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑃)
5		evl1sca.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6		eqid 2731	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	3, 4, 5, 6	ply1sclf 22199	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃))
8	2, 7	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃))
9		ffvelcdm 7014	. . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃))
10	8, 9	sylancom 588	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃))
11		evl1sca.o	. . . 4 ⊢ 𝑂 = (eval₁‘𝑅)
12		eqid 2731	. . . 4 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
13		eqid 2731	. . . 4 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
14	3, 6	ply1bas 22107	. . . 4 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
15	11, 12, 5, 13, 14	evl1val 22244	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐴‘𝑋) ∈ (Base‘𝑃)) → (𝑂‘(𝐴‘𝑋)) = (((1_o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
16	10, 15	syldan 591	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1_o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
17	3, 4	ply1ascl 22172	. . . . . . 7 ⊢ 𝐴 = (algSc‘(1_o mPoly 𝑅))
18	5	ressid 17155	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
20	19	oveq2d 7362	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly 𝑅))
21	20	fveq2d 6826	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly 𝑅)))
22	17, 21	eqtr4id 2785	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))))
23	22	fveq1d 6824	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋))
24	23	fveq2d 6826	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘(𝐴‘𝑋)) = ((1_o eval 𝑅)‘((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋)))
25	12, 5	evlval 22030	. . . . 5 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
26		eqid 2731	. . . . 5 ⊢ (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly (𝑅 ↾_s 𝐵))
27		eqid 2731	. . . . 5 ⊢ (𝑅 ↾_s 𝐵) = (𝑅 ↾_s 𝐵)
28		eqid 2731	. . . . 5 ⊢ (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))
29		1on 8397	. . . . . 6 ⊢ 1_o ∈ On
30	29	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 1_o ∈ On)
31		simpl 482	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ CRing)
32	5	subrgid 20488	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
33	2, 32	syl 17	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
34		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
35	25, 26, 27, 5, 28, 30, 31, 33, 34	evlssca 22024	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋)) = ((𝐵 ↑_m 1_o) × {𝑋}))
36	24, 35	eqtrd 2766	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑_m 1_o) × {𝑋}))
37	36	coeq1d 5800	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((1_o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
38		df1o2 8392	. . . . . . 7 ⊢ 1_o = {∅}
39	5	fvexi 6836	. . . . . . 7 ⊢ 𝐵 ∈ V
40		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
41		eqid 2731	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))
42	38, 39, 40, 41	mapsnf1o3 8819	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o)
43		f1of 6763	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o) → (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵⟶(𝐵 ↑_m 1_o))
44	42, 43	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵⟶(𝐵 ↑_m 1_o))
45	41	fmpt 7043	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (1_o × {𝑦}) ∈ (𝐵 ↑_m 1_o) ↔ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵⟶(𝐵 ↑_m 1_o))
46	44, 45	sylibr 234	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (1_o × {𝑦}) ∈ (𝐵 ↑_m 1_o))
47		eqidd 2732	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})))
48		fconstmpt 5676	. . . . 5 ⊢ ((𝐵 ↑_m 1_o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑_m 1_o) ↦ 𝑋)
49	48	a1i 11	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((𝐵 ↑_m 1_o) × {𝑋}) = (𝑥 ∈ (𝐵 ↑_m 1_o) ↦ 𝑋))
50		eqidd 2732	. . . 4 ⊢ (𝑥 = (1_o × {𝑦}) → 𝑋 = 𝑋)
51	46, 47, 49, 50	fmptcof 7063	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
52		fconstmpt 5676	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
53	51, 52	eqtr4di 2784	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝐵 × {𝑋}))
54	16, 37, 53	3eqtrd 2770	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∅c0 4280 {csn 4573 ↦ cmpt 5170 × cxp 5612 ∘ ccom 5618 Oncon0 6306 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 1_oc1o 8378 ↑_m cmap 8750 Basecbs 17120 ↾_s cress 17141 Ringcrg 20151 CRingccrg 20152 SubRingcsubrg 20484 algSccascl 21789 mPoly cmpl 21843 eval cevl 22008 Poly₁cpl1 22089 eval₁ce1 22229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-rhm 20390 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-lsp 20905 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-ply1 22094 df-evl1 22231

This theorem is referenced by: evl1scad 22250 pf1const 22261 pf1ind 22270 evl1scvarpw 22278 ply1rem 26098 fta1g 26102 fta1blem 26103 plypf1 26144

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