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Mirrors > Home > MPE Home > Th. List > subrgply1 | Structured version Visualization version GIF version |
Description: A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
subrgply1.s | ⊢ 𝑆 = (Poly1‘𝑅) |
subrgply1.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
subrgply1.u | ⊢ 𝑈 = (Poly1‘𝐻) |
subrgply1.b | ⊢ 𝐵 = (Base‘𝑈) |
Ref | Expression |
---|---|
subrgply1 | ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8298 | . . 3 ⊢ 1o ∈ On | |
2 | eqid 2740 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | subrgply1.h | . . . 4 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
4 | eqid 2740 | . . . 4 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
5 | subrgply1.u | . . . . 5 ⊢ 𝑈 = (Poly1‘𝐻) | |
6 | eqid 2740 | . . . . 5 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
7 | subrgply1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
8 | 5, 6, 7 | ply1bas 21362 | . . . 4 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
9 | 2, 3, 4, 8 | subrgmpl 21229 | . . 3 ⊢ ((1o ∈ On ∧ 𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘(1o mPoly 𝑅))) |
10 | 1, 9 | mpan 687 | . 2 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘(1o mPoly 𝑅))) |
11 | eqidd 2741 | . . 3 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘𝑆)) | |
12 | subrgply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
13 | eqid 2740 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
14 | eqid 2740 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
15 | 12, 13, 14 | ply1bas 21362 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1o mPoly 𝑅)) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (Base‘𝑆) = (Base‘(1o mPoly 𝑅))) |
17 | eqid 2740 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
18 | 12, 2, 17 | ply1plusg 21392 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (+g‘𝑆) = (+g‘(1o mPoly 𝑅))) |
20 | 19 | oveqdr 7297 | . . 3 ⊢ ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘(1o mPoly 𝑅))𝑦)) |
21 | eqid 2740 | . . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
22 | 12, 2, 21 | ply1mulr 21394 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘(1o mPoly 𝑅)) |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑆) = (.r‘(1o mPoly 𝑅))) |
24 | 23 | oveqdr 7297 | . . 3 ⊢ ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘(1o mPoly 𝑅))𝑦)) |
25 | 11, 16, 20, 24 | subrgpropd 20055 | . 2 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = (SubRing‘(1o mPoly 𝑅))) |
26 | 10, 25 | eleqtrrd 2844 | 1 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Oncon0 6264 ‘cfv 6431 (class class class)co 7269 1oc1o 8279 Basecbs 16908 ↾s cress 16937 +gcplusg 16958 .rcmulr 16959 SubRingcsubrg 20016 mPoly cmpl 21105 PwSer1cps1 21342 Poly1cpl1 21344 |
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 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
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 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-fz 13237 df-fzo 13380 df-seq 13718 df-hash 14041 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-tset 16977 df-ple 16978 df-0g 17148 df-gsum 17149 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mhm 18426 df-submnd 18427 df-grp 18576 df-minusg 18577 df-mulg 18697 df-subg 18748 df-ghm 18828 df-cntz 18919 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-subrg 20018 df-psr 21108 df-mpl 21110 df-opsr 21112 df-psr1 21347 df-ply1 21349 |
This theorem is referenced by: gsumply1subr 21401 plypf1 25369 |
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