evls1var - Metamath Proof Explorer

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Theorem evls1var 22264

Description: Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)

**Hypotheses**
Ref	Expression
evls1var.q	⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
evls1var.x	⊢ 𝑋 = (var₁‘𝑈)
evls1var.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evls1var.b	⊢ 𝐵 = (Base‘𝑆)
evls1var.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1var.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))

**Assertion**
Ref	Expression
evls1var	⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵))

Proof of Theorem evls1var Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1var.x	. . . 4 ⊢ 𝑋 = (var₁‘𝑈)
2		eqid 2725	. . . . 5 ⊢ (var₁‘𝑆) = (var₁‘𝑆)
3		evls1var.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
4		evls1var.u	. . . . 5 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
5	2, 3, 4	subrgvr1 22187	. . . 4 ⊢ (𝜑 → (var₁‘𝑆) = (var₁‘𝑈))
6	1, 5	eqtr4id 2784	. . 3 ⊢ (𝜑 → 𝑋 = (var₁‘𝑆))
7	6	fveq2d 6894	. 2 ⊢ (𝜑 → (𝑄‘𝑋) = (𝑄‘(var₁‘𝑆)))
8		eqid 2725	. . . . . 6 ⊢ ((1_o evalSub 𝑆)‘𝑅) = ((1_o evalSub 𝑆)‘𝑅)
9		eqid 2725	. . . . . 6 ⊢ (1_o eval 𝑆) = (1_o eval 𝑆)
10		eqid 2725	. . . . . 6 ⊢ (1_o mVar 𝑈) = (1_o mVar 𝑈)
11		evls1var.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
12		1on 8495	. . . . . . 7 ⊢ 1_o ∈ On
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 1_o ∈ On)
14		evls1var.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
15		0lt1o 8521	. . . . . . 7 ⊢ ∅ ∈ 1_o
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1_o)
17	8, 9, 10, 4, 11, 13, 14, 3, 16	evlsvarsrng 22050	. . . . 5 ⊢ (𝜑 → (((1_o evalSub 𝑆)‘𝑅)‘((1_o mVar 𝑈)‘∅)) = ((1_o eval 𝑆)‘((1_o mVar 𝑈)‘∅)))
18	2	vr1val 22117	. . . . . . 7 ⊢ (var₁‘𝑆) = ((1_o mVar 𝑆)‘∅)
19		eqid 2725	. . . . . . . . 9 ⊢ (1_o mVar 𝑆) = (1_o mVar 𝑆)
20	19, 13, 3, 4	subrgmvr 21976	. . . . . . . 8 ⊢ (𝜑 → (1_o mVar 𝑆) = (1_o mVar 𝑈))
21	20	fveq1d 6892	. . . . . . 7 ⊢ (𝜑 → ((1_o mVar 𝑆)‘∅) = ((1_o mVar 𝑈)‘∅))
22	18, 21	eqtrid 2777	. . . . . 6 ⊢ (𝜑 → (var₁‘𝑆) = ((1_o mVar 𝑈)‘∅))
23	22	fveq2d 6894	. . . . 5 ⊢ (𝜑 → (((1_o evalSub 𝑆)‘𝑅)‘(var₁‘𝑆)) = (((1_o evalSub 𝑆)‘𝑅)‘((1_o mVar 𝑈)‘∅)))
24	22	fveq2d 6894	. . . . 5 ⊢ (𝜑 → ((1_o eval 𝑆)‘(var₁‘𝑆)) = ((1_o eval 𝑆)‘((1_o mVar 𝑈)‘∅)))
25	17, 23, 24	3eqtr4d 2775	. . . 4 ⊢ (𝜑 → (((1_o evalSub 𝑆)‘𝑅)‘(var₁‘𝑆)) = ((1_o eval 𝑆)‘(var₁‘𝑆)))
26	25	coeq1d 5856	. . 3 ⊢ (𝜑 → ((((1_o evalSub 𝑆)‘𝑅)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (((1_o eval 𝑆)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
27		eqid 2725	. . . . 5 ⊢ (Poly₁‘𝑈) = (Poly₁‘𝑈)
28		eqid 2725	. . . . . . 7 ⊢ (Poly₁‘(𝑆 ↾_s 𝑅)) = (Poly₁‘(𝑆 ↾_s 𝑅))
29	4	fveq2i 6893	. . . . . . . 8 ⊢ (Poly₁‘𝑈) = (Poly₁‘(𝑆 ↾_s 𝑅))
30	29	fveq2i 6893	. . . . . . 7 ⊢ (Base‘(Poly₁‘𝑈)) = (Base‘(Poly₁‘(𝑆 ↾_s 𝑅)))
31	28, 30	ply1bas 22120	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑈)) = (Base‘(1_o mPoly (𝑆 ↾_s 𝑅)))
32	31	eqcomi 2734	. . . . 5 ⊢ (Base‘(1_o mPoly (𝑆 ↾_s 𝑅))) = (Base‘(Poly₁‘𝑈))
33	2, 3, 4, 27, 32	subrgvr1cl 22188	. . . 4 ⊢ (𝜑 → (var₁‘𝑆) ∈ (Base‘(1_o mPoly (𝑆 ↾_s 𝑅))))
34		evls1var.q	. . . . 5 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
35		eqid 2725	. . . . 5 ⊢ (1_o evalSub 𝑆) = (1_o evalSub 𝑆)
36		eqid 2725	. . . . 5 ⊢ (1_o mPoly (𝑆 ↾_s 𝑅)) = (1_o mPoly (𝑆 ↾_s 𝑅))
37		eqid 2725	. . . . 5 ⊢ (Base‘(1_o mPoly (𝑆 ↾_s 𝑅))) = (Base‘(1_o mPoly (𝑆 ↾_s 𝑅)))
38	34, 35, 11, 36, 37	evls1val 22246	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ (var₁‘𝑆) ∈ (Base‘(1_o mPoly (𝑆 ↾_s 𝑅)))) → (𝑄‘(var₁‘𝑆)) = ((((1_o evalSub 𝑆)‘𝑅)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
39	14, 3, 33, 38	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑄‘(var₁‘𝑆)) = ((((1_o evalSub 𝑆)‘𝑅)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
40		crngring 20187	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
41		eqid 2725	. . . . . 6 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
42		eqid 2725	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
43	41, 42	ply1bas 22120	. . . . . . 7 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(1_o mPoly 𝑆))
44	43	eqcomi 2734	. . . . . 6 ⊢ (Base‘(1_o mPoly 𝑆)) = (Base‘(Poly₁‘𝑆))
45	2, 41, 44	vr1cl 22143	. . . . 5 ⊢ (𝑆 ∈ Ring → (var₁‘𝑆) ∈ (Base‘(1_o mPoly 𝑆)))
46	14, 40, 45	3syl 18	. . . 4 ⊢ (𝜑 → (var₁‘𝑆) ∈ (Base‘(1_o mPoly 𝑆)))
47		eqid 2725	. . . . 5 ⊢ (eval₁‘𝑆) = (eval₁‘𝑆)
48		eqid 2725	. . . . 5 ⊢ (1_o mPoly 𝑆) = (1_o mPoly 𝑆)
49		eqid 2725	. . . . 5 ⊢ (Base‘(1_o mPoly 𝑆)) = (Base‘(1_o mPoly 𝑆))
50	47, 9, 11, 48, 49	evl1val 22255	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ (var₁‘𝑆) ∈ (Base‘(1_o mPoly 𝑆))) → ((eval₁‘𝑆)‘(var₁‘𝑆)) = (((1_o eval 𝑆)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
51	14, 46, 50	syl2anc 582	. . 3 ⊢ (𝜑 → ((eval₁‘𝑆)‘(var₁‘𝑆)) = (((1_o eval 𝑆)‘(var₁‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
52	26, 39, 51	3eqtr4d 2775	. 2 ⊢ (𝜑 → (𝑄‘(var₁‘𝑆)) = ((eval₁‘𝑆)‘(var₁‘𝑆)))
53	47, 2, 11	evl1var 22262	. . 3 ⊢ (𝑆 ∈ CRing → ((eval₁‘𝑆)‘(var₁‘𝑆)) = ( I ↾ 𝐵))
54	14, 53	syl 17	. 2 ⊢ (𝜑 → ((eval₁‘𝑆)‘(var₁‘𝑆)) = ( I ↾ 𝐵))
55	7, 52, 54	3eqtrd 2769	1 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4316 {csn 4622 ↦ cmpt 5224 I cid 5567 × cxp 5668 ↾ cres 5672 ∘ ccom 5674 Oncon0 6362 ‘cfv 6541 (class class class)co 7414 1_oc1o 8476 Basecbs 17177 ↾_s cress 17206 Ringcrg 20175 CRingccrg 20176 SubRingcsubrg 20508 mVar cmvr 21840 mPoly cmpl 21841 evalSub ces 22021 eval cevl 22022 var₁cv1 22101 Poly₁cpl1 22102 evalSub₁ ces1 22239 eval₁ce1 22240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-assa 21789 df-asp 21790 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-evls 22023 df-evl 22024 df-psr1 22105 df-vr1 22106 df-ply1 22107 df-evls1 22241 df-evl1 22242

This theorem is referenced by: evls1varsrng 22266 evls1varpwval 22294 algextdeglem4 33417

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