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Theorem evl1sca 22312

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 20220	. . . . . 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 22263	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐴:𝐵⟶(Base‘𝑃))
8	2, 7	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴:𝐵⟶(Base‘𝑃))
9		ffvelcdm 7028	. . . 4 ⊢ ((𝐴:𝐵⟶(Base‘𝑃) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ (Base‘𝑃))
10	8, 9	sylancom 589	. . 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 22171	. . . 4 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
15	11, 12, 5, 13, 14	evl1val 22307	. . 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 592	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (((1_o eval 𝑅)‘(𝐴‘𝑋)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
17	3, 4	ply1ascl 22236	. . . . . . 7 ⊢ 𝐴 = (algSc‘(1_o mPoly 𝑅))
18	5	ressid 17208	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
20	19	oveq2d 7377	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (1_o mPoly (𝑅 ↾_s 𝐵)) = (1_o mPoly 𝑅))
21	20	fveq2d 6839	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_o mPoly 𝑅)))
22	17, 21	eqtr4id 2791	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝐴 = (algSc‘(1_o mPoly (𝑅 ↾_s 𝐵))))
23	22	fveq1d 6837	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) = ((algSc‘(1_o mPoly (𝑅 ↾_s 𝐵)))‘𝑋))
24	23	fveq2d 6839	. . . 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 22091	. . . . 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 8411	. . . . . 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 20544	. . . . . 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 22085	. . . 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 2772	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((1_o eval 𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑_m 1_o) × {𝑋}))
37	36	coeq1d 5811	. 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 8406	. . . . . . 7 ⊢ 1_o = {∅}
39	5	fvexi 6849	. . . . . . 7 ⊢ 𝐵 ∈ V
40		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
41		eqid 2737	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))
42	38, 39, 40, 41	mapsnf1o3 8837	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑_m 1_o)
43		f1of 6775	. . . . . 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 7057	. . . . 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 5687	. . . . 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 7078	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
52		fconstmpt 5687	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
53	51, 52	eqtr4di 2790	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (((𝐵 ↑_m 1_o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝐵 × {𝑋}))
54	16, 37, 53	3eqtrd 2776	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∅c0 4274 {csn 4568 ↦ cmpt 5167 × cxp 5623 ∘ ccom 5629 Oncon0 6318 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7361 1_oc1o 8392 ↑_m cmap 8767 Basecbs 17173 ↾_s cress 17194 Ringcrg 20208 CRingccrg 20209 SubRingcsubrg 20540 algSccascl 21845 mPoly cmpl 21899 eval cevl 22064 Poly₁cpl1 22153 eval₁ce1 22292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-ofr 7626 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-fz 13456 df-fzo 13603 df-seq 13958 df-hash 14287 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-hom 17238 df-cco 17239 df-0g 17398 df-gsum 17399 df-prds 17404 df-pws 17406 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19038 df-subg 19093 df-ghm 19182 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-srg 20162 df-ring 20210 df-cring 20211 df-rhm 20446 df-subrng 20517 df-subrg 20541 df-lmod 20851 df-lss 20921 df-lsp 20961 df-assa 21846 df-asp 21847 df-ascl 21848 df-psr 21902 df-mvr 21903 df-mpl 21904 df-opsr 21906 df-evls 22065 df-evl 22066 df-psr1 22156 df-ply1 22158 df-evl1 22294

This theorem is referenced by: evl1scad 22313 pf1const 22324 pf1ind 22333 evl1scvarpw 22341 ply1rem 26144 fta1g 26148 fta1blem 26149 plypf1 26190

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