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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 2736 | . . . . . . 7 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | pf1f.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | evlval 21503 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
5 | 4 | rneqi 5892 | . . . . 5 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵) |
6 | 1, 5 | eqtri 2764 | . . . 4 ⊢ 𝐸 = ran ((1o evalSub 𝑅)‘𝐵) |
7 | 6 | mpfrcl 21493 | . . 3 ⊢ (𝐹 ∈ 𝐸 → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅))) |
8 | 7 | simp2d 1143 | . 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing) |
9 | id 22 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸) | |
10 | 9, 1 | eleqtrdi 2848 | . . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1o eval 𝑅)) |
11 | 1on 8423 | . . . . 5 ⊢ 1o ∈ On | |
12 | eqid 2736 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
13 | eqid 2736 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
14 | 2, 3, 12, 13 | evlrhm 21504 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
15 | 11, 8, 14 | sylancr 587 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
16 | eqid 2736 | . . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
17 | eqid 2736 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
18 | eqid 2736 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
19 | 16, 17, 18 | ply1bas 21564 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
20 | eqid 2736 | . . . . 5 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) | |
21 | 19, 20 | rhmf 20156 | . . . 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 6668 | . . . 4 ⊢ ((1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))) → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅))) | |
23 | fvelrnb 6903 | . . . 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 231 | . 2 ⊢ (𝐹 ∈ 𝐸 → ∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹) |
26 | eqid 2736 | . . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
27 | 26, 2, 3, 12, 19 | evl1val 21693 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) = (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
28 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
29 | 26, 16, 28, 3 | evl1rhm 21696 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
30 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
31 | 18, 30 | rhmf 20156 | . . . . . . . 8 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
32 | ffn 6668 | . . . . . . . 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 7031 | . . . . . . 7 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ ran (eval1‘𝑅)) | |
35 | 33, 34 | sylan 580 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ ran (eval1‘𝑅)) |
36 | pf1rcl.q | . . . . . 6 ⊢ 𝑄 = ran (eval1‘𝑅) | |
37 | 35, 36 | eleqtrrdi 2849 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ 𝑄) |
38 | 27, 37 | eqeltrrd 2839 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄) |
39 | coeq1 5813 | . . . . 5 ⊢ (((1o eval 𝑅)‘𝑥) = 𝐹 → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
40 | 39 | eleq1d 2822 | . . . 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 244 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)) |
42 | 41 | rexlimdva 3152 | . 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 Vcvv 3445 {csn 4586 ↦ cmpt 5188 × cxp 5631 ran crn 5634 ∘ ccom 5637 Oncon0 6317 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 1oc1o 8404 ↑m cmap 8764 Basecbs 17082 ↑s cpws 17327 CRingccrg 19963 RingHom crh 20141 SubRingcsubrg 20216 mPoly cmpl 21306 evalSub ces 21478 eval cevl 21479 PwSer1cps1 21544 Poly1cpl1 21546 eval1ce1 21678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-ofr 7617 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-fzo 13567 df-seq 13906 df-hash 14230 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-hom 17156 df-cco 17157 df-0g 17322 df-gsum 17323 df-prds 17328 df-pws 17330 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-mulg 18871 df-subg 18923 df-ghm 19004 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-srg 19916 df-ring 19964 df-cring 19965 df-rnghom 20144 df-subrg 20218 df-lmod 20322 df-lss 20391 df-lsp 20431 df-assa 21257 df-asp 21258 df-ascl 21259 df-psr 21309 df-mvr 21310 df-mpl 21311 df-opsr 21313 df-evls 21480 df-evl 21481 df-psr1 21549 df-ply1 21551 df-evl1 21680 |
This theorem is referenced by: pf1ind 21719 |
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