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Mirrors > Home > MPE Home > Th. List > pf1subrg | Structured version Visualization version GIF version |
Description: Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
pf1const.b | ⊢ 𝐵 = (Base‘𝑅) |
pf1const.q | ⊢ 𝑄 = ran (eval1‘𝑅) |
Ref | Expression |
---|---|
pf1subrg | ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅 ↑s 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2738 | . . . . 5 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
3 | eqid 2738 | . . . . 5 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
4 | pf1const.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 1, 2, 3, 4 | evl1rhm 21486 | . . . 4 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
6 | eqid 2738 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
7 | eqid 2738 | . . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
8 | 6, 7 | rhmf 19958 | . . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
9 | ffn 6593 | . . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) | |
10 | fnima 6556 | . . . 4 ⊢ ((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) → ((eval1‘𝑅) “ (Base‘(Poly1‘𝑅))) = ran (eval1‘𝑅)) | |
11 | 5, 8, 9, 10 | 4syl 19 | . . 3 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅) “ (Base‘(Poly1‘𝑅))) = ran (eval1‘𝑅)) |
12 | pf1const.q | . . 3 ⊢ 𝑄 = ran (eval1‘𝑅) | |
13 | 11, 12 | eqtr4di 2796 | . 2 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅) “ (Base‘(Poly1‘𝑅))) = 𝑄) |
14 | 2 | ply1assa 21358 | . . . 4 ⊢ (𝑅 ∈ CRing → (Poly1‘𝑅) ∈ AssAlg) |
15 | assaring 21056 | . . . 4 ⊢ ((Poly1‘𝑅) ∈ AssAlg → (Poly1‘𝑅) ∈ Ring) | |
16 | 6 | subrgid 20014 | . . . 4 ⊢ ((Poly1‘𝑅) ∈ Ring → (Base‘(Poly1‘𝑅)) ∈ (SubRing‘(Poly1‘𝑅))) |
17 | 14, 15, 16 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ CRing → (Base‘(Poly1‘𝑅)) ∈ (SubRing‘(Poly1‘𝑅))) |
18 | rhmima 20043 | . . 3 ⊢ (((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) ∧ (Base‘(Poly1‘𝑅)) ∈ (SubRing‘(Poly1‘𝑅))) → ((eval1‘𝑅) “ (Base‘(Poly1‘𝑅))) ∈ (SubRing‘(𝑅 ↑s 𝐵))) | |
19 | 5, 17, 18 | syl2anc 584 | . 2 ⊢ (𝑅 ∈ CRing → ((eval1‘𝑅) “ (Base‘(Poly1‘𝑅))) ∈ (SubRing‘(𝑅 ↑s 𝐵))) |
20 | 13, 19 | eqeltrrd 2840 | 1 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅 ↑s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ran crn 5586 “ cima 5588 Fn wfn 6422 ⟶wf 6423 ‘cfv 6427 (class class class)co 7268 Basecbs 16900 ↑s cpws 17145 Ringcrg 19771 CRingccrg 19772 RingHom crh 19944 SubRingcsubrg 20008 AssAlgcasa 21045 Poly1cpl1 21336 eval1ce1 21468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-pm 8606 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-sup 9189 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-fz 13228 df-fzo 13371 df-seq 13710 df-hash 14033 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-hom 16974 df-cco 16975 df-0g 17140 df-gsum 17141 df-prds 17146 df-pws 17148 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-submnd 18419 df-grp 18568 df-minusg 18569 df-sbg 18570 df-mulg 18689 df-subg 18740 df-ghm 18820 df-cntz 18911 df-cmn 19376 df-abl 19377 df-mgp 19709 df-ur 19726 df-srg 19730 df-ring 19773 df-cring 19774 df-rnghom 19947 df-subrg 20010 df-lmod 20113 df-lss 20182 df-lsp 20222 df-assa 21048 df-asp 21049 df-ascl 21050 df-psr 21100 df-mvr 21101 df-mpl 21102 df-opsr 21104 df-evls 21270 df-evl 21271 df-psr1 21339 df-ply1 21341 df-evl1 21470 |
This theorem is referenced by: pf1f 21504 pf1addcl 21507 pf1mulcl 21508 |
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