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Theorem evl1sca 22359

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 20272	. . . . . 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 2740	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	3, 4, 5, 6	ply1sclf 22309	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃))
8	2, 7	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃))
9		ffvelcdm 7115	. . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃))
10	8, 9	sylancom 587	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃))
11		evl1sca.o	. . . 4 ⊢ 𝑂 = (eval₁‘𝑅)
12		eqid 2740	. . . 4 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
13		eqid 2740	. . . 4 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
14	3, 6	ply1bas 22217	. . . 4 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
15	11, 12, 5, 13, 14	evl1val 22354	. . 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 590	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1_o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
17	3, 4	ply1ascl 22282	. . . . . . 7 ⊢ 𝐴 = (algSc‘(1_o mPoly 𝑅))
18	5	ressid 17303	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
20	19	oveq2d 7464	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly 𝑅))
21	20	fveq2d 6924	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly 𝑅)))
22	17, 21	eqtr4id 2799	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))))
23	22	fveq1d 6922	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋))
24	23	fveq2d 6924	. . . 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 22142	. . . . 5 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
26		eqid 2740	. . . . 5 ⊢ (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly (𝑅 ↾_s 𝐵))
27		eqid 2740	. . . . 5 ⊢ (𝑅 ↾_s 𝐵) = (𝑅 ↾_s 𝐵)
28		eqid 2740	. . . . 5 ⊢ (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))
29		1on 8534	. . . . . 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 20601	. . . . . 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 22136	. . . 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 2780	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑_m 1_o) × {𝑋}))
37	36	coeq1d 5886	. 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 8529	. . . . . . 7 ⊢ 1_o = {∅}
39	5	fvexi 6934	. . . . . . 7 ⊢ 𝐵 ∈ V
40		0ex 5325	. . . . . . 7 ⊢ ∅ ∈ V
41		eqid 2740	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))
42	38, 39, 40, 41	mapsnf1o3 8953	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o)
43		f1of 6862	. . . . . 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 7144	. . . . 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 2741	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})))
48		fconstmpt 5762	. . . . 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 2741	. . . 4 ⊢ (𝑥 = (1_o × {𝑦}) → 𝑋 = 𝑋)
51	46, 47, 49, 50	fmptcof 7164	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
52		fconstmpt 5762	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
53	51, 52	eqtr4di 2798	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝐵 × {𝑋}))
54	16, 37, 53	3eqtrd 2784	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∅c0 4352 {csn 4648 ↦ cmpt 5249 × cxp 5698 ∘ ccom 5704 Oncon0 6395 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 1_oc1o 8515 ↑_m cmap 8884 Basecbs 17258 ↾_s cress 17287 Ringcrg 20260 CRingccrg 20261 SubRingcsubrg 20595 algSccascl 21895 mPoly cmpl 21949 eval cevl 22120 Poly₁cpl1 22199 eval₁ce1 22339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-ply1 22204 df-evl1 22341

This theorem is referenced by: evl1scad 22360 pf1const 22371 pf1ind 22380 evl1scvarpw 22388 ply1rem 26225 fta1g 26229 fta1blem 26230 plypf1 26271

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