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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 2820 | . . . . . . 7 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | pf1f.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | evlval 20304 | . . . . . 6 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵) |
5 | 4 | rneqi 5804 | . . . . 5 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵) |
6 | 1, 5 | eqtri 2843 | . . . 4 ⊢ 𝐸 = ran ((1o evalSub 𝑅)‘𝐵) |
7 | 6 | mpfrcl 20294 | . . 3 ⊢ (𝐹 ∈ 𝐸 → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅))) |
8 | 7 | simp2d 1138 | . 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing) |
9 | id 22 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸) | |
10 | 9, 1 | eleqtrdi 2922 | . . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1o eval 𝑅)) |
11 | 1on 8106 | . . . . 5 ⊢ 1o ∈ On | |
12 | eqid 2820 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
13 | eqid 2820 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
14 | 2, 3, 12, 13 | evlrhm 20305 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
15 | 11, 8, 14 | sylancr 589 | . . . 4 ⊢ (𝐹 ∈ 𝐸 → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
16 | eqid 2820 | . . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
17 | eqid 2820 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
18 | eqid 2820 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
19 | 16, 17, 18 | ply1bas 20359 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
20 | eqid 2820 | . . . . 5 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) | |
21 | 19, 20 | rhmf 19474 | . . . 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 6511 | . . . 4 ⊢ ((1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))) → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅))) | |
23 | fvelrnb 6723 | . . . 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 234 | . 2 ⊢ (𝐹 ∈ 𝐸 → ∃𝑥 ∈ (Base‘(Poly1‘𝑅))((1o eval 𝑅)‘𝑥) = 𝐹) |
26 | eqid 2820 | . . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
27 | 26, 2, 3, 12, 19 | evl1val 20488 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) = (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
28 | eqid 2820 | . . . . . . . . 9 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
29 | 26, 16, 28, 3 | evl1rhm 20491 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
30 | eqid 2820 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
31 | 18, 30 | rhmf 19474 | . . . . . . . 8 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
32 | ffn 6511 | . . . . . . . 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 6845 | . . . . . . 7 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ ran (eval1‘𝑅)) | |
35 | 33, 34 | sylan 582 | . . . . . 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 2923 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑥) ∈ 𝑄) |
38 | 27, 37 | eqeltrrd 2913 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄) |
39 | coeq1 5725 | . . . . 5 ⊢ (((1o eval 𝑅)‘𝑥) = 𝐹 → (((1o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
40 | 39 | eleq1d 2896 | . . . 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 247 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)) |
42 | 41 | rexlimdva 3283 | . 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 Vcvv 3493 {csn 4564 ↦ cmpt 5143 × cxp 5550 ran crn 5553 ∘ ccom 5556 Oncon0 6188 Fn wfn 6347 ⟶wf 6348 ‘cfv 6352 (class class class)co 7153 1oc1o 8092 ↑m cmap 8403 Basecbs 16479 ↑s cpws 16716 CRingccrg 19294 RingHom crh 19460 SubRingcsubrg 19527 mPoly cmpl 20129 evalSub ces 20280 eval cevl 20281 PwSer1cps1 20339 Poly1cpl1 20341 eval1ce1 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-of 7406 df-ofr 7407 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-map 8405 df-pm 8406 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-sup 8903 df-oi 8971 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-fzo 13032 df-seq 13368 df-hash 13689 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-hom 16585 df-cco 16586 df-0g 16711 df-gsum 16712 df-prds 16717 df-pws 16719 df-mre 16853 df-mrc 16854 df-acs 16856 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-mhm 17952 df-submnd 17953 df-grp 18102 df-minusg 18103 df-sbg 18104 df-mulg 18221 df-subg 18272 df-ghm 18352 df-cntz 18443 df-cmn 18904 df-abl 18905 df-mgp 19236 df-ur 19248 df-srg 19252 df-ring 19295 df-cring 19296 df-rnghom 19463 df-subrg 19529 df-lmod 19632 df-lss 19700 df-lsp 19740 df-assa 20081 df-asp 20082 df-ascl 20083 df-psr 20132 df-mvr 20133 df-mpl 20134 df-opsr 20136 df-evls 20282 df-evl 20283 df-psr1 20344 df-ply1 20346 df-evl1 20475 |
This theorem is referenced by: pf1ind 20514 |
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