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Mirrors > Home > MPE Home > Th. List > ressply1evl | Structured version Visualization version GIF version |
Description: Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.) |
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)) |
Ref | Expression |
---|---|
ressply1evl | ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵)) |
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 Σg (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 Σg (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 Σg 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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