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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 22351 | . . . . . 6 ⊢ 𝐸 = (𝑆 evalSub1 𝐾) |
4 | eqid 2735 | . . . . . 6 ⊢ (Poly1‘(𝑆 ↾s 𝐾)) = (Poly1‘(𝑆 ↾s 𝐾)) | |
5 | eqid 2735 | . . . . . 6 ⊢ (𝑆 ↾s 𝐾) = (𝑆 ↾s 𝐾) | |
6 | eqid 2735 | . . . . . 6 ⊢ (Base‘(Poly1‘(𝑆 ↾s 𝐾))) = (Base‘(Poly1‘(𝑆 ↾s 𝐾))) | |
7 | ressply1evl.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑆 ∈ CRing) |
9 | 7 | crngringd 20264 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring) |
10 | 2 | subrgid 20590 | . . . . . . . 8 ⊢ (𝑆 ∈ Ring → 𝐾 ∈ (SubRing‘𝑆)) |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘𝑆)) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝐾 ∈ (SubRing‘𝑆)) |
13 | eqid 2735 | . . . . . . . . . 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 2735 | . . . . . . . . . 10 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
19 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈)) | |
20 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆)) | |
21 | 13, 14, 15, 16, 17, 18, 19, 20 | ressply1bas2 22245 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆)))) |
22 | inss2 4246 | . . . . . . . . 9 ⊢ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆))) ⊆ (Base‘(Poly1‘𝑆)) | |
23 | 21, 22 | eqsstrdi 4050 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly1‘𝑆))) |
24 | 2 | ressid 17290 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐾) = 𝑆) |
25 | 7, 24 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ↾s 𝐾) = 𝑆) |
26 | 25 | fveq2d 6911 | . . . . . . . . 9 ⊢ (𝜑 → (Poly1‘(𝑆 ↾s 𝐾)) = (Poly1‘𝑆)) |
27 | 26 | fveq2d 6911 | . . . . . . . 8 ⊢ (𝜑 → (Base‘(Poly1‘(𝑆 ↾s 𝐾))) = (Base‘(Poly1‘𝑆))) |
28 | 23, 27 | sseqtrrd 4037 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly1‘(𝑆 ↾s 𝐾)))) |
29 | 28 | sselda 3995 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ (Base‘(Poly1‘(𝑆 ↾s 𝐾)))) |
30 | eqid 2735 | . . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
31 | eqid 2735 | . . . . . 6 ⊢ (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆)) | |
32 | eqid 2735 | . . . . . 6 ⊢ (coe1‘𝑚) = (coe1‘𝑚) | |
33 | 3, 2, 4, 5, 6, 8, 12, 29, 30, 31, 32 | evls1fpws 22389 | . . . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ (SubRing‘𝑆)) |
36 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵) | |
37 | 34, 2, 15, 14, 16, 8, 35, 36, 30, 31, 32 | evls1fpws 22389 | . . . . 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 2778 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐵) → (𝐸‘𝑚) = (𝑄‘𝑚)) |
39 | 38 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝐵 (𝐸‘𝑚) = (𝑄‘𝑚)) |
40 | eqid 2735 | . . . . . . 7 ⊢ (𝑆 ↑s 𝐾) = (𝑆 ↑s 𝐾) | |
41 | 1, 13, 40, 2 | evl1rhm 22352 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝐸 ∈ ((Poly1‘𝑆) RingHom (𝑆 ↑s 𝐾))) |
42 | eqid 2735 | . . . . . . 7 ⊢ (Base‘(𝑆 ↑s 𝐾)) = (Base‘(𝑆 ↑s 𝐾)) | |
43 | 20, 42 | rhmf 20502 | . . . . . 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 6738 | . . . 4 ⊢ (𝜑 → 𝐸 Fn (Base‘(Poly1‘𝑆))) |
46 | 34, 2, 40, 14, 15 | evls1rhm 22342 | . . . . . . 7 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾))) |
47 | 7, 17, 46 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾))) |
48 | 16, 42 | rhmf 20502 | . . . . . 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 6738 | . . . 4 ⊢ (𝜑 → 𝑄 Fn 𝐵) |
51 | fvreseq1 7059 | . . . 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 838 | . . 3 ⊢ (𝜑 → ((𝐸 ↾ 𝐵) = 𝑄 ↔ ∀𝑚 ∈ 𝐵 (𝐸‘𝑚) = (𝑄‘𝑚))) |
53 | 39, 52 | mpbird 257 | . 2 ⊢ (𝜑 → (𝐸 ↾ 𝐵) = 𝑄) |
54 | 53 | eqcomd 2741 | 1 ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 ↦ cmpt 5231 ↾ cres 5691 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Σg cgsu 17487 ↑s cpws 17493 .gcmg 19098 mulGrpcmgp 20152 Ringcrg 20251 CRingccrg 20252 RingHom crh 20486 SubRingcsubrg 20586 PwSer1cps1 22192 Poly1cpl1 22194 coe1cco1 22195 evalSub1 ces1 22333 eval1ce1 22334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 |
This theorem is referenced by: evls1addd 22391 evls1muld 22392 evls1vsca 22393 evls1fvcl 22395 evls1maprhm 22396 evls1subd 33577 irngss 33702 rtelextdg2lem 33732 |
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