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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 2795 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | subrgmvr.v | . . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2795 | . . . 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 19235 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | 1, 2, 3, 4, 7 | mvrf 19897 | . . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅))) |
9 | 8 | ffnd 6388 | . 2 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
10 | subrgmvr.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | 2, 4, 5, 10 | subrgmvr 19934 | . . . . . 6 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
12 | 11 | fveq1d 6545 | . . . . 5 ⊢ (𝜑 → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
14 | subrgmvrf.u | . . . . 5 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
15 | eqid 2795 | . . . . 5 ⊢ (𝐼 mVar 𝐻) = (𝐼 mVar 𝐻) | |
16 | subrgmvrf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
17 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
18 | 10 | subrgring 19233 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
19 | 5, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
20 | 19 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 ∈ Ring) |
21 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
22 | 14, 15, 16, 17, 20, 21 | mvrcl 19922 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 mVar 𝐻)‘𝑥) ∈ 𝐵) |
23 | 13, 22 | eqeltrd 2883 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐵) |
24 | 23 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵) |
25 | ffnfv 6750 | . 2 ⊢ (𝑉:𝐼⟶𝐵 ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵)) | |
26 | 9, 24, 25 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 Basecbs 16317 ↾s cress 16318 Ringcrg 18992 SubRingcsubrg 19226 mPwSer cmps 19824 mVar cmvr 19825 mPoly cmpl 19826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-tset 16418 df-0g 16549 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-grp 17869 df-subg 18035 df-mgp 18935 df-ur 18947 df-ring 18994 df-subrg 19228 df-psr 19829 df-mvr 19830 df-mpl 19831 |
This theorem is referenced by: subrgvr1cl 20118 |
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