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Mirrors > Home > MPE Home > Th. List > mpfpf1 | Structured version Visualization version GIF version |
Description: Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
pf1rcl.q | ⊢ 𝑄 = ran (eval1‘𝑅) |
pf1f.b | ⊢ 𝐵 = (Base‘𝑅) |
mpfpf1.q | ⊢ 𝐸 = ran (1o eval 𝑅) |
Ref | Expression |
---|---|
mpfpf1 | ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpfpf1.q | . . . . 5 ⊢ 𝐸 = ran (1o eval 𝑅) | |
2 | eqid 2773 | . . . . . . 7 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | pf1f.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | evlval 20030 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
5 | 4 | rneqi 5648 | . . . . 5 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵) |
6 | 1, 5 | eqtri 2797 | . . . 4 ⊢ 𝐸 = ran ((1o evalSub 𝑅)‘𝐵) |
7 | 6 | mpfrcl 20024 | . . 3 ⊢ (𝐹 ∈ 𝐸 → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅))) |
8 | 7 | simp2d 1124 | . 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing) |
9 | id 22 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸) | |
10 | 9, 1 | syl6eleq 2871 | . . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1o eval 𝑅)) |
11 | 1on 7911 | . . . . 5 ⊢ 1o ∈ On | |
12 | eqid 2773 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
13 | eqid 2773 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑𝑚 1o)) = (𝑅 ↑s (𝐵 ↑𝑚 1o)) | |
14 | 2, 3, 12, 13 | evlrhm 20031 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1o)))) |
15 | 11, 8, 14 | sylancr 579 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1o)))) |
16 | eqid 2773 | . . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
17 | eqid 2773 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
18 | eqid 2773 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
19 | 16, 17, 18 | ply1bas 20082 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
20 | eqid 2773 | . . . . 5 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1o))) | |
21 | 19, 20 | rhmf 19214 | . . . 4 ⊢ ((1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1o))) → (1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1o)))) |
22 | ffn 6342 | . . . 4 ⊢ ((1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1o))) → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅))) | |
23 | fvelrnb 6554 | . . . 4 ⊢ ((1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)) → (𝐹 ∈ ran (1o eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹)) | |
24 | 15, 21, 22, 23 | 4syl 19 | . . 3 ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∈ ran (1o eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹)) |
25 | 10, 24 | mpbid 224 | . 2 ⊢ (𝐹 ∈ 𝐸 → ∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹) |
26 | eqid 2773 | . . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
27 | 26, 2, 3, 12, 19 | evl1val 20210 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) = (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
28 | eqid 2773 | . . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
29 | 26, 16, 28, 3 | evl1rhm 20213 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
30 | eqid 2773 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
31 | 18, 30 | rhmf 19214 | . . . . . . . 8 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
32 | ffn 6342 | . . . . . . . 8 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) | |
33 | 29, 31, 32 | 3syl 18 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) |
34 | fnfvelrn 6672 | . . . . . . 7 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ ran (eval1‘𝑅)) | |
35 | 33, 34 | sylan 572 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ ran (eval1‘𝑅)) |
36 | pf1rcl.q | . . . . . 6 ⊢ 𝑄 = ran (eval1‘𝑅) | |
37 | 35, 36 | syl6eleqr 2872 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ 𝑄) |
38 | 27, 37 | eqeltrrd 2862 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄) |
39 | coeq1 5575 | . . . . 5 ⊢ (((1o eval 𝑅)‘𝑥) = 𝐹 → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
40 | 39 | eleq1d 2845 | . . . 4 ⊢ (((1o eval 𝑅)‘𝑥) = 𝐹 → ((((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄 ↔ (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)) |
41 | 38, 40 | syl5ibcom 237 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)) |
42 | 41 | rexlimdva 3224 | . 2 ⊢ (𝑅 ∈ CRing → (∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)) |
43 | 8, 25, 42 | sylc 65 | 1 ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∃wrex 3084 Vcvv 3410 {csn 4436 ↦ cmpt 5005 × cxp 5402 ran crn 5405 ∘ ccom 5408 Oncon0 6027 Fn wfn 6181 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 1oc1o 7897 ↑𝑚 cmap 8205 Basecbs 16338 ↑s cpws 16575 CRingccrg 19034 RingHom crh 19200 SubRingcsubrg 19267 mPoly cmpl 19860 evalSub ces 20010 eval cevl 20011 PwSer1cps1 20062 Poly1cpl1 20064 eval1ce1 20196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-ofr 7227 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-ixp 8259 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-fsupp 8628 df-sup 8700 df-oi 8768 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-uz 12058 df-fz 12708 df-fzo 12849 df-seq 13184 df-hash 13505 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-sca 16436 df-vsca 16437 df-ip 16438 df-tset 16439 df-ple 16440 df-ds 16442 df-hom 16444 df-cco 16445 df-0g 16570 df-gsum 16571 df-prds 16576 df-pws 16578 df-mre 16728 df-mrc 16729 df-acs 16731 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-mhm 17816 df-submnd 17817 df-grp 17907 df-minusg 17908 df-sbg 17909 df-mulg 18025 df-subg 18073 df-ghm 18140 df-cntz 18231 df-cmn 18681 df-abl 18682 df-mgp 18976 df-ur 18988 df-srg 18992 df-ring 19035 df-cring 19036 df-rnghom 19203 df-subrg 19269 df-lmod 19371 df-lss 19439 df-lsp 19479 df-assa 19819 df-asp 19820 df-ascl 19821 df-psr 19863 df-mvr 19864 df-mpl 19865 df-opsr 19867 df-evls 20012 df-evl 20013 df-psr1 20067 df-ply1 20069 df-evl1 20198 |
This theorem is referenced by: pf1ind 20236 |
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