Theorem evls1var 22479

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 2763	. . . . 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 22403	. . . 4 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈))
6	1, 5	eqtr4id 2817	. . 3 ⊢ (𝜑 → 𝑋 = (var1‘𝑆))
7	6	fveq2d 6887	. 2 ⊢ (𝜑 → (𝑄‘𝑋) = (𝑄‘(var1‘𝑆)))
8		eqid 2763	. . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
9		eqid 2763	. . . . . 6 ⊢ (1o eval 𝑆) = (1o eval 𝑆)
10		eqid 2763	. . . . . 6 ⊢ (1o mVar 𝑈) = (1o mVar 𝑈)
11		evls1var.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
12		1on 8467	. . . . . . 7 ⊢ 1o ∈ On
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ On)
14		evls1var.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
15		0lt1o 8490	. . . . . . 7 ⊢ ∅ ∈ 1o
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
17	8, 9, 10, 4, 11, 13, 14, 3, 16	evlsvarsrng 22239	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅)))
18	2	vr1val 22333	. . . . . . 7 ⊢ (var1‘𝑆) = ((1o mVar 𝑆)‘∅)
19		eqid 2763	. . . . . . . . 9 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
20	19, 13, 3, 4	subrgmvr 22165	. . . . . . . 8 ⊢ (𝜑 → (1o mVar 𝑆) = (1o mVar 𝑈))
21	20	fveq1d 6885	. . . . . . 7 ⊢ (𝜑 → ((1o mVar 𝑆)‘∅) = ((1o mVar 𝑈)‘∅))
22	18, 21	eqtrid 2810	. . . . . 6 ⊢ (𝜑 → (var1‘𝑆) = ((1o mVar 𝑈)‘∅))
23	22	fveq2d 6887	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)))
24	22	fveq2d 6887	. . . . 5 ⊢ (𝜑 → ((1o eval 𝑆)‘(var1‘𝑆)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅)))
25	17, 23, 24	3eqtr4d 2808	. . . 4 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = ((1o eval 𝑆)‘(var1‘𝑆)))
26	25	coeq1d 5849	. . 3 ⊢ (𝜑 → ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
27		eqid 2763	. . . . 5 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈)
28		eqid 2763	. . . . . . 7 ⊢ (Poly1‘(𝑆 ↾s 𝑅)) = (Poly1‘(𝑆 ↾s 𝑅))
29	4	fveq2i 6886	. . . . . . . 8 ⊢ (Poly1‘𝑈) = (Poly1‘(𝑆 ↾s 𝑅))
30	29	fveq2i 6886	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘(𝑆 ↾s 𝑅)))
31	28, 30	ply1bas 22336	. . . . . 6 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(1o mPoly (𝑆 ↾s 𝑅)))
32	31	eqcomi 2772	. . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(Poly1‘𝑈))
33	2, 3, 4, 27, 32	subrgvr1cl 22404	. . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅))))
34		evls1var.q	. . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
35		eqid 2763	. . . . 5 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆)
36		eqid 2763	. . . . 5 ⊢ (1o mPoly (𝑆 ↾s 𝑅)) = (1o mPoly (𝑆 ↾s 𝑅))
37		eqid 2763	. . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(1o mPoly (𝑆 ↾s 𝑅)))
38	34, 35, 11, 36, 37	evls1val 22461	. . . 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 1398	. . 3 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
40		crngring 20328	. . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
41		eqid 2763	. . . . . 6 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆)
42		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆))
43	41, 42	ply1bas 22336	. . . . . . 7 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(1o mPoly 𝑆))
44	43	eqcomi 2772	. . . . . 6 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(Poly1‘𝑆))
45	2, 41, 44	vr1cl 22358	. . . . 5 ⊢ (𝑆 ∈ Ring → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆)))
46	14, 40, 45	3syl 19	. . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆)))
47		eqid 2763	. . . . 5 ⊢ (eval1‘𝑆) = (eval1‘𝑆)
48		eqid 2763	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
49		eqid 2763	. . . . 5 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(1o mPoly 𝑆))
50	47, 9, 11, 48, 49	evl1val 22470	. . . 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 595	. . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
52	26, 39, 51	3eqtr4d 2808	. 2 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((eval1‘𝑆)‘(var1‘𝑆)))
53	47, 2, 11	evl1var 22477	. . 3 ⊢ (𝑆 ∈ CRing → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵))
54	14, 53	syl 18	. 2 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵))
55	7, 52, 54	3eqtrd 2802	1 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∅c0 4287  {csn 4590   ↦ cmpt 5193   I cid 5557   × cxp 5661   ↾ cres 5665   ∘ ccom 5667  Oncon0 6362  ‘cfv 6538  (class class class)co 7412  1_oc1o 8447  Basecbs 17270   ↾_s cress 17291  Ringcrg 20316  CRingccrg 20317  SubRingcsubrg 20655   mVar cmvr 22036   mPoly cmpl 22037   evalSub ces 22204   eval cevl 22205  var₁cv1 22317  Poly₁cpl1 22318   evalSub₁ ces1 22454  eval₁ce1 22455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-ofr 7677  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17495  df-gsum 17496  df-prds 17501  df-pws 17503  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-cntz 19388  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-srg 20270  df-ring 20318  df-cring 20319  df-rhm 20555  df-subrng 20632  df-subrg 20656  df-lmod 20964  df-lss 21034  df-lsp 21074  df-assa 21984  df-asp 21985  df-ascl 21986  df-psr 22040  df-mvr 22041  df-mpl 22042  df-opsr 22044  df-evls 22206  df-evl 22207  df-psr1 22321  df-vr1 22322  df-ply1 22323  df-evls1 22456  df-evl1 22457

This theorem is referenced by:  evls1varsrng  22481  evls1varpwval  22509  vr1nz  33864  algextdeglem4  34091  2sqr3minply  34151  cos9thpiminplylem6  34158