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Mirrors > Home > MPE Home > Th. List > mpfsubrg | Structured version Visualization version GIF version |
Description: Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by AV, 19-Sep-2021.) |
Ref | Expression |
---|---|
mpfsubrg.q | ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) |
Ref | Expression |
---|---|
mpfsubrg | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpfsubrg.q | . . 3 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | eqid 2734 | . . . . 5 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅) | |
3 | eqid 2734 | . . . . 5 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅)) | |
4 | eqid 2734 | . . . . 5 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅) | |
5 | eqid 2734 | . . . . 5 ⊢ (𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)) = (𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)) | |
6 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 2, 3, 4, 5, 6 | evlsrhm 22129 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) |
8 | eqid 2734 | . . . . 5 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
9 | eqid 2734 | . . . . 5 ⊢ (Base‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼))) = (Base‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼))) | |
10 | 8, 9 | rhmf 20501 | . . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) |
11 | ffn 6736 | . . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) | |
12 | fnima 6698 | . . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅) “ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = ran ((𝐼 evalSub 𝑆)‘𝑅)) | |
13 | 7, 10, 11, 12 | 4syl 19 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (((𝐼 evalSub 𝑆)‘𝑅) “ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = ran ((𝐼 evalSub 𝑆)‘𝑅)) |
14 | 1, 13 | eqtr4id 2793 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (((𝐼 evalSub 𝑆)‘𝑅) “ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))) |
15 | 4 | subrgring 20590 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝑅) ∈ Ring) |
16 | 3 | mplring 22056 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑆 ↾s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
17 | 15, 16 | sylan2 593 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
18 | 17 | 3adant2 1130 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring) |
19 | 8 | subrgid 20589 | . . . 4 ⊢ ((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring → (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ (SubRing‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ (SubRing‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
21 | rhmima 20620 | . . 3 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑m 𝐼))) ∧ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ (SubRing‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅) “ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) | |
22 | 7, 20, 21 | syl2anc 584 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (((𝐼 evalSub 𝑆)‘𝑅) “ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) |
23 | 14, 22 | eqeltrd 2838 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ran crn 5689 “ cima 5691 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 Basecbs 17244 ↾s cress 17273 ↑s cpws 17492 Ringcrg 20250 CRingccrg 20251 RingHom crh 20485 SubRingcsubrg 20585 mPoly cmpl 21943 evalSub ces 22113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-srg 20204 df-ring 20252 df-cring 20253 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-lmod 20876 df-lss 20947 df-lsp 20987 df-assa 21890 df-asp 21891 df-ascl 21892 df-psr 21946 df-mvr 21947 df-mpl 21948 df-evls 22115 |
This theorem is referenced by: mpff 22145 mpfaddcl 22146 mpfmulcl 22147 |
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