Theorem evls1var 22223

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 2729	. . . . 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 22145	. . . 4 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈))
6	1, 5	eqtr4id 2783	. . 3 ⊢ (𝜑 → 𝑋 = (var1‘𝑆))
7	6	fveq2d 6826	. 2 ⊢ (𝜑 → (𝑄‘𝑋) = (𝑄‘(var1‘𝑆)))
8		eqid 2729	. . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
9		eqid 2729	. . . . . 6 ⊢ (1o eval 𝑆) = (1o eval 𝑆)
10		eqid 2729	. . . . . 6 ⊢ (1o mVar 𝑈) = (1o mVar 𝑈)
11		evls1var.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
12		1on 8400	. . . . . . 7 ⊢ 1o ∈ On
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ On)
14		evls1var.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
15		0lt1o 8422	. . . . . . 7 ⊢ ∅ ∈ 1o
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
17	8, 9, 10, 4, 11, 13, 14, 3, 16	evlsvarsrng 22004	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅)))
18	2	vr1val 22074	. . . . . . 7 ⊢ (var1‘𝑆) = ((1o mVar 𝑆)‘∅)
19		eqid 2729	. . . . . . . . 9 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
20	19, 13, 3, 4	subrgmvr 21938	. . . . . . . 8 ⊢ (𝜑 → (1o mVar 𝑆) = (1o mVar 𝑈))
21	20	fveq1d 6824	. . . . . . 7 ⊢ (𝜑 → ((1o mVar 𝑆)‘∅) = ((1o mVar 𝑈)‘∅))
22	18, 21	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → (var1‘𝑆) = ((1o mVar 𝑈)‘∅))
23	22	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)))
24	22	fveq2d 6826	. . . . 5 ⊢ (𝜑 → ((1o eval 𝑆)‘(var1‘𝑆)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅)))
25	17, 23, 24	3eqtr4d 2774	. . . 4 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = ((1o eval 𝑆)‘(var1‘𝑆)))
26	25	coeq1d 5804	. . 3 ⊢ (𝜑 → ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
27		eqid 2729	. . . . 5 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
28		eqid 2729	. . . . . . 7 ⊢ (Poly1‘(𝑆 ↾s 𝑅)) = (Poly1‘(𝑆 ↾s 𝑅))
29	4	fveq2i 6825	. . . . . . . 8 ⊢ (Poly1‘𝑈) = (Poly1‘(𝑆 ↾s 𝑅))
30	29	fveq2i 6825	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘(𝑆 ↾s 𝑅)))
31	28, 30	ply1bas 22077	. . . . . 6 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(1o mPoly (𝑆 ↾s 𝑅)))
32	31	eqcomi 2738	. . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(Poly1‘𝑈))
33	2, 3, 4, 27, 32	subrgvr1cl 22146	. . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅))))
34		evls1var.q	. . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
35		eqid 2729	. . . . 5 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆)
36		eqid 2729	. . . . 5 ⊢ (1o mPoly (𝑆 ↾s 𝑅)) = (1o mPoly (𝑆 ↾s 𝑅))
37		eqid 2729	. . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(1o mPoly (𝑆 ↾s 𝑅)))
38	34, 35, 11, 36, 37	evls1val 22205	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅)))) → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
39	14, 3, 33, 38	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
40		crngring 20130	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
41		eqid 2729	. . . . . 6 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆)
42		eqid 2729	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆))
43	41, 42	ply1bas 22077	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(1o mPoly 𝑆))
44	43	eqcomi 2738	. . . . . 6 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(Poly1‘𝑆))
45	2, 41, 44	vr1cl 22100	. . . . 5 ⊢ (𝑆 ∈ Ring → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆)))
46	14, 40, 45	3syl 18	. . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆)))
47		eqid 2729	. . . . 5 ⊢ (eval1‘𝑆) = (eval1‘𝑆)
48		eqid 2729	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
49		eqid 2729	. . . . 5 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(1o mPoly 𝑆))
50	47, 9, 11, 48, 49	evl1val 22214	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
51	14, 46, 50	syl2anc 584	. . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
52	26, 39, 51	3eqtr4d 2774	. 2 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((eval1‘𝑆)‘(var1‘𝑆)))
53	47, 2, 11	evl1var 22221	. . 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 2768	1 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4284  {csn 4577   ↦ cmpt 5173   I cid 5513   × cxp 5617   ↾ cres 5621   ∘ ccom 5623  Oncon0 6307  ‘cfv 6482  (class class class)co 7349  1_oc1o 8381  Basecbs 17120   ↾_s cress 17141  Ringcrg 20118  CRingccrg 20119  SubRingcsubrg 20454   mVar cmvr 21812   mPoly cmpl 21813   evalSub ces 21977   eval cevl 21978  var₁cv1 22058  Poly₁cpl1 22059   evalSub₁ ces1 22198  eval₁ce1 22199

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-srg 20072  df-ring 20120  df-cring 20121  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-lmod 20765  df-lss 20835  df-lsp 20875  df-assa 21760  df-asp 21761  df-ascl 21762  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-evls 21979  df-evl 21980  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-evls1 22200  df-evl1 22201

This theorem is referenced by:  evls1varsrng  22225  evls1varpwval  22253  vr1nz  33526  algextdeglem4  33687  2sqr3minply  33747  cos9thpiminplylem6  33754