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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 22254 | . . . 4 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) |
5 | 4 | fveq1i 6903 | . . 3 ⊢ (𝑂‘𝐴) = (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)‘𝐴) |
6 | 1on 8505 | . . . . . 6 ⊢ 1o ∈ On | |
7 | simpl 481 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ CRing) | |
8 | evl1val.m | . . . . . . 7 ⊢ 𝑀 = (1o mPoly 𝑅) | |
9 | eqid 2728 | . . . . . . 7 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
10 | 2, 3, 8, 9 | evlrhm 22049 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑀 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
11 | 6, 7, 10 | sylancr 585 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑄 ∈ (𝑀 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | evl1val.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑀) | |
13 | eqid 2728 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) | |
14 | 12, 13 | rhmf 20431 | . . . . 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 7002 | . . . 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 586 | . . 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 2780 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘(𝑄‘𝐴))) |
19 | ffvelcdm 7096 | . . . . 5 ⊢ ((𝑄:𝐾⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
20 | 15, 19 | sylancom 586 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
21 | crngring 20192 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
22 | 21 | adantr 479 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ Ring) |
23 | ovex 7459 | . . . . 5 ⊢ (𝐵 ↑m 1o) ∈ V | |
24 | 9, 3 | pwsbas 17476 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐵 ↑m 1o) ∈ V) → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
25 | 22, 23, 24 | sylancl 584 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
26 | 20, 25 | eleqtrrd 2832 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) ∈ (𝐵 ↑m (𝐵 ↑m 1o))) |
27 | coeq1 5864 | . . . 4 ⊢ (𝑥 = (𝑄‘𝐴) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
28 | eqid 2728 | . . . 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 6915 | . . . . 5 ⊢ (𝑄‘𝐴) ∈ V | |
30 | 3 | fvexi 6916 | . . . . . 6 ⊢ 𝐵 ∈ V |
31 | 30 | mptex 7241 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) ∈ V |
32 | 29, 31 | coex 7944 | . . . 4 ⊢ ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ V |
33 | 27, 28, 32 | fvmpt 7010 | . . 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 2768 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ↦ cmpt 5235 × cxp 5680 ∘ ccom 5686 Oncon0 6374 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 1oc1o 8486 ↑m cmap 8851 Basecbs 17187 ↑s cpws 17435 Ringcrg 20180 CRingccrg 20181 RingHom crh 20415 mPoly cmpl 21846 eval cevl 22024 eval1ce1 22240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-evls 22025 df-evl 22026 df-evl1 22242 |
This theorem is referenced by: evl1sca 22260 evl1var 22262 evls1var 22264 mpfpf1 22277 pf1mpf 22278 pf1ind 22281 |
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