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Mirrors > Home > MPE Home > Th. List > evl1val | Structured version Visualization version GIF version |
Description: Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
evl1fval.o | ⊢ 𝑂 = (eval1‘𝑅) |
evl1fval.q | ⊢ 𝑄 = (1o eval 𝑅) |
evl1fval.b | ⊢ 𝐵 = (Base‘𝑅) |
evl1val.m | ⊢ 𝑀 = (1o mPoly 𝑅) |
evl1val.k | ⊢ 𝐾 = (Base‘𝑀) |
Ref | Expression |
---|---|
evl1val | ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1fval.o | . . . . 5 ⊢ 𝑂 = (eval1‘𝑅) | |
2 | evl1fval.q | . . . . 5 ⊢ 𝑄 = (1o eval 𝑅) | |
3 | evl1fval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 21577 | . . . 4 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) |
5 | 4 | fveq1i 6813 | . . 3 ⊢ (𝑂‘𝐴) = (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)‘𝐴) |
6 | 1on 8358 | . . . . . 6 ⊢ 1o ∈ On | |
7 | simpl 483 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ CRing) | |
8 | evl1val.m | . . . . . . 7 ⊢ 𝑀 = (1o mPoly 𝑅) | |
9 | eqid 2737 | . . . . . . 7 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
10 | 2, 3, 8, 9 | evlrhm 21389 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑀 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
11 | 6, 7, 10 | sylancr 587 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑄 ∈ (𝑀 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | evl1val.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑀) | |
13 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) | |
14 | 12, 13 | rhmf 20045 | . . . . 5 ⊢ (𝑄 ∈ (𝑀 RingHom (𝑅 ↑s (𝐵 ↑m 1o))) → 𝑄:𝐾⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
15 | 11, 14 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑄:𝐾⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
16 | fvco3 6907 | . . . 4 ⊢ ((𝑄:𝐾⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ∧ 𝐴 ∈ 𝐾) → (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)‘𝐴) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴))) | |
17 | 15, 16 | sylancom 588 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)‘𝐴) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴))) |
18 | 5, 17 | eqtrid 2789 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴))) |
19 | ffvelcdm 6999 | . . . . 5 ⊢ ((𝑄:𝐾⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
20 | 15, 19 | sylancom 588 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
21 | crngring 19870 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring) |
23 | ovex 7350 | . . . . 5 ⊢ (𝐵 ↑m 1o) ∈ V | |
24 | 9, 3 | pwsbas 17275 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐵 ↑m 1o) ∈ V) → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
25 | 22, 23, 24 | sylancl 586 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
26 | 20, 25 | eleqtrrd 2841 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (𝐵 ↑m (𝐵 ↑m 1o))) |
27 | coeq1 5787 | . . . 4 ⊢ (𝑥 = (𝑄‘𝐴) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
28 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
29 | fvex 6825 | . . . . 5 ⊢ (𝑄‘𝐴) ∈ V | |
30 | 3 | fvexi 6826 | . . . . . 6 ⊢ 𝐵 ∈ V |
31 | 30 | mptex 7139 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) ∈ V |
32 | 29, 31 | coex 7824 | . . . 4 ⊢ ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ V |
33 | 27, 28, 32 | fvmpt 6915 | . . 3 ⊢ ((𝑄‘𝐴) ∈ (𝐵 ↑m (𝐵 ↑m 1o)) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴)) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
34 | 26, 33 | syl 17 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴)) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
35 | 18, 34 | eqtrd 2777 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 {csn 4571 ↦ cmpt 5170 × cxp 5606 ∘ ccom 5612 Oncon0 6289 ⟶wf 6462 ‘cfv 6466 (class class class)co 7317 1oc1o 8339 ↑m cmap 8665 Basecbs 16989 ↑s cpws 17234 Ringcrg 19858 CRingccrg 19859 RingHom crh 20031 mPoly cmpl 21192 eval cevl 21364 eval1ce1 21563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630 ax-cnex 11007 ax-resscn 11008 ax-1cn 11009 ax-icn 11010 ax-addcl 11011 ax-addrcl 11012 ax-mulcl 11013 ax-mulrcl 11014 ax-mulcom 11015 ax-addass 11016 ax-mulass 11017 ax-distr 11018 ax-i2m1 11019 ax-1ne0 11020 ax-1rid 11021 ax-rnegex 11022 ax-rrecex 11023 ax-cnre 11024 ax-pre-lttri 11025 ax-pre-lttrn 11026 ax-pre-ltadd 11027 ax-pre-mulgt0 11028 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-iin 4940 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5563 df-se 5564 df-we 5565 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-pred 6225 df-ord 6292 df-on 6293 df-lim 6294 df-suc 6295 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-isom 6475 df-riota 7274 df-ov 7320 df-oprab 7321 df-mpo 7322 df-of 7575 df-ofr 7576 df-om 7760 df-1st 7878 df-2nd 7879 df-supp 8027 df-frecs 8146 df-wrecs 8177 df-recs 8251 df-rdg 8290 df-1o 8346 df-er 8548 df-map 8667 df-pm 8668 df-ixp 8736 df-en 8784 df-dom 8785 df-sdom 8786 df-fin 8787 df-fsupp 9206 df-sup 9278 df-oi 9346 df-card 9775 df-pnf 11091 df-mnf 11092 df-xr 11093 df-ltxr 11094 df-le 11095 df-sub 11287 df-neg 11288 df-nn 12054 df-2 12116 df-3 12117 df-4 12118 df-5 12119 df-6 12120 df-7 12121 df-8 12122 df-9 12123 df-n0 12314 df-z 12400 df-dec 12518 df-uz 12663 df-fz 13320 df-fzo 13463 df-seq 13802 df-hash 14125 df-struct 16925 df-sets 16942 df-slot 16960 df-ndx 16972 df-base 16990 df-ress 17019 df-plusg 17052 df-mulr 17053 df-sca 17055 df-vsca 17056 df-ip 17057 df-tset 17058 df-ple 17059 df-ds 17061 df-hom 17063 df-cco 17064 df-0g 17229 df-gsum 17230 df-prds 17235 df-pws 17237 df-mre 17372 df-mrc 17373 df-acs 17375 df-mgm 18403 df-sgrp 18452 df-mnd 18463 df-mhm 18507 df-submnd 18508 df-grp 18656 df-minusg 18657 df-sbg 18658 df-mulg 18777 df-subg 18828 df-ghm 18908 df-cntz 18999 df-cmn 19463 df-abl 19464 df-mgp 19796 df-ur 19813 df-srg 19817 df-ring 19860 df-cring 19861 df-rnghom 20034 df-subrg 20104 df-lmod 20208 df-lss 20277 df-lsp 20317 df-assa 21143 df-asp 21144 df-ascl 21145 df-psr 21195 df-mvr 21196 df-mpl 21197 df-evls 21365 df-evl 21366 df-evl1 21565 |
This theorem is referenced by: evl1sca 21583 evl1var 21585 evls1var 21587 mpfpf1 21600 pf1mpf 21601 pf1ind 21604 |
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