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Mirrors > Home > MPE Home > Th. List > pf1id | Structured version Visualization version GIF version |
Description: The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
pf1const.b | ⊢ 𝐵 = (Base‘𝑅) |
pf1const.q | ⊢ 𝑄 = ran (eval1‘𝑅) |
Ref | Expression |
---|---|
pf1id | ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2728 | . . . 4 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
3 | pf1const.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1var 22255 | . . 3 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅)‘(var1‘𝑅)) = ( I ↾ 𝐵)) |
5 | eqid 2728 | . . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
6 | eqid 2728 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
7 | 1, 5, 6, 3 | evl1rhm 22251 | . . . . 5 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
8 | eqid 2728 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
9 | eqid 2728 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
10 | 8, 9 | rhmf 20424 | . . . . 5 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
11 | ffn 6722 | . . . . 5 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) | |
12 | 7, 10, 11 | 3syl 18 | . . . 4 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) |
13 | crngring 20185 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
14 | 2, 5, 8 | vr1cl 22136 | . . . . 5 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) |
16 | fnfvelrn 7090 | . . . 4 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘(var1‘𝑅)) ∈ ran (eval1‘𝑅)) | |
17 | 12, 15, 16 | syl2anc 583 | . . 3 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅)‘(var1‘𝑅)) ∈ ran (eval1‘𝑅)) |
18 | 4, 17 | eqeltrrd 2830 | . 2 ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ ran (eval1‘𝑅)) |
19 | pf1const.q | . 2 ⊢ 𝑄 = ran (eval1‘𝑅) | |
20 | 18, 19 | eleqtrrdi 2840 | 1 ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 I cid 5575 ran crn 5679 ↾ cres 5680 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 ↑s cpws 17428 Ringcrg 20173 CRingccrg 20174 RingHom crh 20408 var1cv1 22095 Poly1cpl1 22096 eval1ce1 22233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-assa 21787 df-asp 21788 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-opsr 21846 df-evls 22018 df-evl 22019 df-psr1 22099 df-vr1 22100 df-ply1 22101 df-evl1 22235 |
This theorem is referenced by: (None) |
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