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Theorem ressply1evl 22296

Description: Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.)

**Hypotheses**
Ref	Expression
ressply1evl2.q	⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
ressply1evl2.k	⊢ 𝐾 = (Base‘𝑆)
ressply1evl2.w	⊢ 𝑊 = (Poly₁‘𝑈)
ressply1evl2.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
ressply1evl2.b	⊢ 𝐵 = (Base‘𝑊)
ressply1evl.e	⊢ 𝐸 = (eval₁‘𝑆)
ressply1evl.s	⊢ (𝜑 → 𝑆 ∈ CRing)
ressply1evl.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))

**Assertion**
Ref	Expression
ressply1evl	⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵))

Proof of Theorem ressply1evl Dummy variables 𝑘 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressply1evl.e	. . . . . . 7 ⊢ 𝐸 = (eval₁‘𝑆)
2		ressply1evl2.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑆)
3	1, 2	evl1fval1 22257	. . . . . 6 ⊢ 𝐸 = (𝑆 evalSub₁ 𝐾)
4		eqid 2728	. . . . . 6 ⊢ (Poly₁‘(𝑆 ↾_s 𝐾)) = (Poly₁‘(𝑆 ↾_s 𝐾))
5		eqid 2728	. . . . . 6 ⊢ (𝑆 ↾_s 𝐾) = (𝑆 ↾_s 𝐾)
6		eqid 2728	. . . . . 6 ⊢ (Base‘(Poly₁‘(𝑆 ↾_s 𝐾))) = (Base‘(Poly₁‘(𝑆 ↾_s 𝐾)))
7		ressply1evl.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
8	7	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑆 ∈ CRing)
9	7	crngringd 20193	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring)
10	2	subrgid 20519	. . . . . . . 8 ⊢ (𝑆 ∈ Ring → 𝐾 ∈ (SubRing‘𝑆))
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘𝑆))
12	11	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝐾 ∈ (SubRing‘𝑆))
13		eqid 2728	. . . . . . . . . 10 ⊢ (Poly₁‘𝑆) = (Poly₁‘𝑆)
14		ressply1evl2.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
15		ressply1evl2.w	. . . . . . . . . 10 ⊢ 𝑊 = (Poly₁‘𝑈)
16		ressply1evl2.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
17		ressply1evl.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
18		eqid 2728	. . . . . . . . . 10 ⊢ (PwSer₁‘𝑈) = (PwSer₁‘𝑈)
19		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘(PwSer₁‘𝑈)) = (Base‘(PwSer₁‘𝑈))
20		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑆)) = (Base‘(Poly₁‘𝑆))
21	13, 14, 15, 16, 17, 18, 19, 20	ressply1bas2 22153	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))))
22		inss2 4232	. . . . . . . . 9 ⊢ ((Base‘(PwSer₁‘𝑈)) ∩ (Base‘(Poly₁‘𝑆))) ⊆ (Base‘(Poly₁‘𝑆))
23	21, 22	eqsstrdi 4036	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘𝑆)))
24	2	ressid 17232	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾_s 𝐾) = 𝑆)
25	7, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾_s 𝐾) = 𝑆)
26	25	fveq2d 6906	. . . . . . . . 9 ⊢ (𝜑 → (Poly₁‘(𝑆 ↾_s 𝐾)) = (Poly₁‘𝑆))
27	26	fveq2d 6906	. . . . . . . 8 ⊢ (𝜑 → (Base‘(Poly₁‘(𝑆 ↾_s 𝐾))) = (Base‘(Poly₁‘𝑆)))
28	23, 27	sseqtrrd 4023	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly₁‘(𝑆 ↾_s 𝐾))))
29	28	sselda 3982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ (Base‘(Poly₁‘(𝑆 ↾_s 𝐾))))
30		eqid 2728	. . . . . 6 ⊢ (._r‘𝑆) = (._r‘𝑆)
31		eqid 2728	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘𝑆))
32		eqid 2728	. . . . . 6 ⊢ (coe₁‘𝑚) = (coe₁‘𝑚)
33	3, 2, 4, 5, 6, 8, 12, 29, 30, 31, 32	evls1fpws 22295	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝐸‘𝑚) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝑚)‘𝑘)(._r‘𝑆)(𝑘(._g‘(mulGrp‘𝑆))𝑥))))))
34		ressply1evl2.q	. . . . . 6 ⊢ 𝑄 = (𝑆 evalSub₁ 𝑅)
35	17	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ (SubRing‘𝑆))
36		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵)
37	34, 2, 15, 14, 16, 8, 35, 36, 30, 31, 32	evls1fpws 22295	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝑄‘𝑚) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σ_g (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝑚)‘𝑘)(._r‘𝑆)(𝑘(._g‘(mulGrp‘𝑆))𝑥))))))
38	33, 37	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝐸‘𝑚) = (𝑄‘𝑚))
39	38	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝐵 (𝐸‘𝑚) = (𝑄‘𝑚))
40		eqid 2728	. . . . . . 7 ⊢ (𝑆 ↑_s 𝐾) = (𝑆 ↑_s 𝐾)
41	1, 13, 40, 2	evl1rhm 22258	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝐸 ∈ ((Poly₁‘𝑆) RingHom (𝑆 ↑_s 𝐾)))
42		eqid 2728	. . . . . . 7 ⊢ (Base‘(𝑆 ↑_s 𝐾)) = (Base‘(𝑆 ↑_s 𝐾))
43	20, 42	rhmf 20431	. . . . . 6 ⊢ (𝐸 ∈ ((Poly₁‘𝑆) RingHom (𝑆 ↑_s 𝐾)) → 𝐸:(Base‘(Poly₁‘𝑆))⟶(Base‘(𝑆 ↑_s 𝐾)))
44	7, 41, 43	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐸:(Base‘(Poly₁‘𝑆))⟶(Base‘(𝑆 ↑_s 𝐾)))
45	44	ffnd 6728	. . . 4 ⊢ (𝜑 → 𝐸 Fn (Base‘(Poly₁‘𝑆)))
46	34, 2, 40, 14, 15	evls1rhm 22248	. . . . . . 7 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑_s 𝐾)))
47	7, 17, 46	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑_s 𝐾)))
48	16, 42	rhmf 20431	. . . . . 6 ⊢ (𝑄 ∈ (𝑊 RingHom (𝑆 ↑_s 𝐾)) → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s 𝐾)))
49	47, 48	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s 𝐾)))
50	49	ffnd 6728	. . . 4 ⊢ (𝜑 → 𝑄 Fn 𝐵)
51		fvreseq1 7053	. . . 4 ⊢ (((𝐸 Fn (Base‘(Poly₁‘𝑆)) ∧ 𝑄 Fn 𝐵) ∧ 𝐵 ⊆ (Base‘(Poly₁‘𝑆))) → ((𝐸 ↾ 𝐵) = 𝑄 ↔ ∀𝑚 ∈ 𝐵 (𝐸‘𝑚) = (𝑄‘𝑚)))
52	45, 50, 23, 51	syl21anc 836	. . 3 ⊢ (𝜑 → ((𝐸 ↾ 𝐵) = 𝑄 ↔ ∀𝑚 ∈ 𝐵 (𝐸‘𝑚) = (𝑄‘𝑚)))
53	39, 52	mpbird 256	. 2 ⊢ (𝜑 → (𝐸 ↾ 𝐵) = 𝑄)
54	53	eqcomd 2734	1 ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∩ cin 3948 ⊆ wss 3949 ↦ cmpt 5235 ↾ cres 5684 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℕ₀cn0 12510 Basecbs 17187 ↾_s cress 17216 ._rcmulr 17241 Σ_g cgsu 17429 ↑_s cpws 17435 ._gcmg 19030 mulGrpcmgp 20081 Ringcrg 20180 CRingccrg 20181 RingHom crh 20415 SubRingcsubrg 20513 PwSer₁cps1 22101 Poly₁cpl1 22103 coe₁cco1 22104 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-evls 22025 df-evl 22026 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-coe1 22109 df-evls1 22241 df-evl1 22242

This theorem is referenced by: evls1addd 22297 evls1muld 22298 evls1vsca 22299 evls1fvcl 22301 evls1maprhm 22302 evls1subd 33289 irngss 33398

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