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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 2740 | . . . 4 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2740 | . . . 4 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
3 | pf1const.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1var 21492 | . . 3 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅)‘(var1‘𝑅)) = ( I ↾ 𝐵)) |
5 | eqid 2740 | . . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
6 | eqid 2740 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
7 | 1, 5, 6, 3 | evl1rhm 21488 | . . . . 5 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
8 | eqid 2740 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
9 | eqid 2740 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
10 | 8, 9 | rhmf 19960 | . . . . 5 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
11 | ffn 6597 | . . . . 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 19785 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
14 | 2, 5, 8 | vr1cl 21378 | . . . . 5 ⊢ (𝑅 ∈ Ring → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) |
16 | fnfvelrn 6953 | . . . 4 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ (var1‘𝑅) ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘(var1‘𝑅)) ∈ ran (eval1‘𝑅)) | |
17 | 12, 15, 16 | syl2anc 584 | . . 3 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅)‘(var1‘𝑅)) ∈ ran (eval1‘𝑅)) |
18 | 4, 17 | eqeltrrd 2842 | . 2 ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ ran (eval1‘𝑅)) |
19 | pf1const.q | . 2 ⊢ 𝑄 = ran (eval1‘𝑅) | |
20 | 18, 19 | eleqtrrdi 2852 | 1 ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 I cid 5488 ran crn 5590 ↾ cres 5591 Fn wfn 6426 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 ↑s cpws 17147 Ringcrg 19773 CRingccrg 19774 RingHom crh 19946 var1cv1 21337 Poly1cpl1 21338 eval1ce1 21470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-ofr 7526 df-om 7702 df-1st 7818 df-2nd 7819 df-supp 7963 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-map 8592 df-pm 8593 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-fsupp 9099 df-sup 9171 df-oi 9239 df-card 9690 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-fz 13231 df-fzo 13374 df-seq 13712 df-hash 14035 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-hom 16976 df-cco 16977 df-0g 17142 df-gsum 17143 df-prds 17148 df-pws 17150 df-mre 17285 df-mrc 17286 df-acs 17288 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-mhm 18420 df-submnd 18421 df-grp 18570 df-minusg 18571 df-sbg 18572 df-mulg 18691 df-subg 18742 df-ghm 18822 df-cntz 18913 df-cmn 19378 df-abl 19379 df-mgp 19711 df-ur 19728 df-srg 19732 df-ring 19775 df-cring 19776 df-rnghom 19949 df-subrg 20012 df-lmod 20115 df-lss 20184 df-lsp 20224 df-assa 21050 df-asp 21051 df-ascl 21052 df-psr 21102 df-mvr 21103 df-mpl 21104 df-opsr 21106 df-evls 21272 df-evl 21273 df-psr1 21341 df-vr1 21342 df-ply1 21343 df-evl1 21472 |
This theorem is referenced by: (None) |
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