![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrgmvrf | Structured version Visualization version GIF version |
Description: The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
subrgmvr.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
subrgmvr.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
subrgmvr.r | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
subrgmvr.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
subrgmvrf.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
subrgmvrf.b | ⊢ 𝐵 = (Base‘𝑈) |
Ref | Expression |
---|---|
subrgmvrf | ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | subrgmvr.v | . . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2725 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | subrgmvr.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | subrgmvr.r | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
6 | subrgrcl 20555 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | 1, 2, 3, 4, 7 | mvrf 21986 | . . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅))) |
9 | 8 | ffnd 6728 | . 2 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
10 | subrgmvr.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 2, 4, 5, 10 | subrgmvr 22032 | . . . . . 6 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
12 | 11 | fveq1d 6902 | . . . . 5 ⊢ (𝜑 → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
13 | 12 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
14 | subrgmvrf.u | . . . . 5 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
15 | eqid 2725 | . . . . 5 ⊢ (𝐼 mVar 𝐻) = (𝐼 mVar 𝐻) | |
16 | subrgmvrf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
17 | 4 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
18 | 10 | subrgring 20553 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
19 | 5, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
20 | 19 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 ∈ Ring) |
21 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
22 | 14, 15, 16, 17, 20, 21 | mvrcl 21993 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 mVar 𝐻)‘𝑥) ∈ 𝐵) |
23 | 13, 22 | eqeltrd 2825 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐵) |
24 | 23 | ralrimiva 3135 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵) |
25 | ffnfv 7132 | . 2 ⊢ (𝑉:𝐼⟶𝐵 ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵)) | |
26 | 9, 24, 25 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 ↾s cress 17237 Ringcrg 20211 SubRingcsubrg 20546 mPwSer cmps 21893 mVar cmvr 21894 mPoly cmpl 21895 |
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 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 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-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-tset 17280 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-subg 19112 df-mgp 20113 df-ur 20160 df-ring 20213 df-subrg 20548 df-psr 21898 df-mvr 21899 df-mpl 21900 |
This theorem is referenced by: subrgvr1cl 22245 |
Copyright terms: Public domain | W3C validator |