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Theorem evl1sca 22338

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 20242	. . . . . 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 2737	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	3, 4, 5, 6	ply1sclf 22288	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃))
8	2, 7	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃))
9		ffvelcdm 7101	. . . 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 2737	. . . 4 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
13		eqid 2737	. . . 4 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
14	3, 6	ply1bas 22196	. . . 4 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
15	11, 12, 5, 13, 14	evl1val 22333	. . 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 22261	. . . . . . 7 ⊢ 𝐴 = (algSc‘(1_o mPoly 𝑅))
18	5	ressid 17290	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
20	19	oveq2d 7447	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly 𝑅))
21	20	fveq2d 6910	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly 𝑅)))
22	17, 21	eqtr4id 2796	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))))
23	22	fveq1d 6908	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋))
24	23	fveq2d 6910	. . . 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 22119	. . . . 5 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
26		eqid 2737	. . . . 5 ⊢ (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly (𝑅 ↾_s 𝐵))
27		eqid 2737	. . . . 5 ⊢ (𝑅 ↾_s 𝐵) = (𝑅 ↾_s 𝐵)
28		eqid 2737	. . . . 5 ⊢ (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))
29		1on 8518	. . . . . 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 20573	. . . . . 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 22113	. . . 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 2777	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑_m 1_o) × {𝑋}))
37	36	coeq1d 5872	. 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 8513	. . . . . . 7 ⊢ 1_o = {∅}
39	5	fvexi 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
40		0ex 5307	. . . . . . 7 ⊢ ∅ ∈ V
41		eqid 2737	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))
42	38, 39, 40, 41	mapsnf1o3 8935	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o)
43		f1of 6848	. . . . . 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 7130	. . . . 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 2738	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})))
48		fconstmpt 5747	. . . . 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 2738	. . . 4 ⊢ (𝑥 = (1_o × {𝑦}) → 𝑋 = 𝑋)
51	46, 47, 49, 50	fmptcof 7150	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
52		fconstmpt 5747	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
53	51, 52	eqtr4di 2795	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝐵 × {𝑋}))
54	16, 37, 53	3eqtrd 2781	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 {csn 4626 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 Oncon0 6384 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1_oc1o 8499 ↑_m cmap 8866 Basecbs 17247 ↾_s cress 17274 Ringcrg 20230 CRingccrg 20231 SubRingcsubrg 20569 algSccascl 21872 mPoly cmpl 21926 eval cevl 22097 Poly₁cpl1 22178 eval₁ce1 22318

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-ply1 22183 df-evl1 22320

This theorem is referenced by: evl1scad 22339 pf1const 22350 pf1ind 22359 evl1scvarpw 22367 ply1rem 26205 fta1g 26209 fta1blem 26210 plypf1 26251

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