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Theorem evl1sca 22255

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 20169	. . . . . 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 22205	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃))
8	2, 7	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃))
9		ffvelcdm 7020	. . . 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 22113	. . . 4 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
15	11, 12, 5, 13, 14	evl1val 22250	. . 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 22178	. . . . . . 7 ⊢ 𝐴 = (algSc‘(1_o mPoly 𝑅))
18	5	ressid 17161	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
20	19	oveq2d 7368	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly 𝑅))
21	20	fveq2d 6832	. . . . . . 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 6830	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋))
24	23	fveq2d 6832	. . . 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 22036	. . . . 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 8403	. . . . . 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 20494	. . . . . 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 22030	. . . 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 5806	. 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 8398	. . . . . . 7 ⊢ 1_o = {∅}
39	5	fvexi 6842	. . . . . . 7 ⊢ 𝐵 ∈ V
40		0ex 5247	. . . . . . 7 ⊢ ∅ ∈ V
41		eqid 2731	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))
42	38, 39, 40, 41	mapsnf1o3 8825	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o)
43		f1of 6769	. . . . . 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 7049	. . . . 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 5681	. . . . 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 7069	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
52		fconstmpt 5681	. . 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 4282 {csn 4575 ↦ cmpt 5174 × cxp 5617 ∘ ccom 5623 Oncon0 6312 ⟶wf 6483 –1-1-onto→wf1o 6486 ‘cfv 6487 (class class class)co 7352 1_oc1o 8384 ↑_m cmap 8756 Basecbs 17126 ↾_s cress 17147 Ringcrg 20157 CRingccrg 20158 SubRingcsubrg 20490 algSccascl 21795 mPoly cmpl 21849 eval cevl 22014 Poly₁cpl1 22095 eval₁ce1 22235

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 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089

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 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-fz 13414 df-fzo 13561 df-seq 13915 df-hash 14244 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-hom 17191 df-cco 17192 df-0g 17351 df-gsum 17352 df-prds 17357 df-pws 17359 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18987 df-subg 19042 df-ghm 19131 df-cntz 19235 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20396 df-subrng 20467 df-subrg 20491 df-lmod 20801 df-lss 20871 df-lsp 20911 df-assa 21796 df-asp 21797 df-ascl 21798 df-psr 21852 df-mvr 21853 df-mpl 21854 df-opsr 21856 df-evls 22015 df-evl 22016 df-psr1 22098 df-ply1 22100 df-evl1 22237

This theorem is referenced by: evl1scad 22256 pf1const 22267 pf1ind 22276 evl1scvarpw 22284 ply1rem 26104 fta1g 26108 fta1blem 26109 plypf1 26150

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