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| Mirrors > Home > MPE Home > Th. List > subrgascl | Structured version Visualization version GIF version | ||
| Description: The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.) | 
| Ref | Expression | 
|---|---|
| subrgascl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) | 
| subrgascl.a | ⊢ 𝐴 = (algSc‘𝑃) | 
| subrgascl.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) | 
| subrgascl.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) | 
| subrgascl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) | 
| subrgascl.r | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | 
| subrgascl.c | ⊢ 𝐶 = (algSc‘𝑈) | 
| Ref | Expression | 
|---|---|
| subrgascl | ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subrgascl.c | . . . 4 ⊢ 𝐶 = (algSc‘𝑈) | |
| 2 | eqid 2736 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
| 3 | eqid 2736 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
| 4 | 1, 2, 3 | asclfn 21902 | . . 3 ⊢ 𝐶 Fn (Base‘(Scalar‘𝑈)) | 
| 5 | subrgascl.r | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
| 6 | subrgascl.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
| 7 | 6 | subrgbas 20582 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) | 
| 8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 = (Base‘𝐻)) | 
| 9 | subrgascl.u | . . . . . . 7 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
| 10 | subrgascl.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
| 11 | 6 | ovexi 7466 | . . . . . . . 8 ⊢ 𝐻 ∈ V | 
| 12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V) | 
| 13 | 9, 10, 12 | mplsca 22034 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈)) | 
| 14 | 13 | fveq2d 6909 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈))) | 
| 15 | 8, 14 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈))) | 
| 16 | 15 | fneq2d 6661 | . . 3 ⊢ (𝜑 → (𝐶 Fn 𝑇 ↔ 𝐶 Fn (Base‘(Scalar‘𝑈)))) | 
| 17 | 4, 16 | mpbiri 258 | . 2 ⊢ (𝜑 → 𝐶 Fn 𝑇) | 
| 18 | subrgascl.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
| 19 | eqid 2736 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
| 20 | eqid 2736 | . . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
| 21 | 18, 19, 20 | asclfn 21902 | . . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃)) | 
| 22 | subrgascl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
| 23 | subrgrcl 20577 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
| 24 | 5, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 25 | 22, 10, 24 | mplsca 22034 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) | 
| 26 | 25 | fveq2d 6909 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) | 
| 27 | 26 | fneq2d 6661 | . . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃)))) | 
| 28 | 21, 27 | mpbiri 258 | . . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅)) | 
| 29 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 30 | 29 | subrgss 20573 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) | 
| 31 | 5, 30 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) | 
| 32 | fnssres 6690 | . . 3 ⊢ ((𝐴 Fn (Base‘𝑅) ∧ 𝑇 ⊆ (Base‘𝑅)) → (𝐴 ↾ 𝑇) Fn 𝑇) | |
| 33 | 28, 31, 32 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝑇) Fn 𝑇) | 
| 34 | fvres 6924 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) | |
| 35 | 34 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) | 
| 36 | eqid 2736 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 37 | 6, 36 | subrg0 20580 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) | 
| 38 | 5, 37 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) | 
| 39 | 38 | ifeq2d 4545 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) | 
| 40 | 39 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) | 
| 41 | 40 | mpteq2dv 5243 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅))) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻)))) | 
| 42 | eqid 2736 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 43 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐼 ∈ 𝑊) | 
| 44 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑅 ∈ Ring) | 
| 45 | 31 | sselda 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅)) | 
| 46 | 22, 42, 36, 29, 18, 43, 44, 45 | mplascl 22089 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) | 
| 47 | eqid 2736 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
| 48 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 49 | 6 | subrgring 20575 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) | 
| 50 | 5, 49 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) | 
| 51 | 50 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐻 ∈ Ring) | 
| 52 | 8 | eleq2d 2826 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ (Base‘𝐻))) | 
| 53 | 52 | biimpa 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝐻)) | 
| 54 | 9, 42, 47, 48, 1, 43, 51, 53 | mplascl 22089 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻)))) | 
| 55 | 41, 46, 54 | 3eqtr4d 2786 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥)) | 
| 56 | 35, 55 | eqtr2d 2777 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥)) | 
| 57 | 17, 33, 56 | eqfnfvd 7053 | 1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 ⊆ wss 3950 ifcif 4524 {csn 4625 ↦ cmpt 5224 × cxp 5682 ◡ccnv 5683 ↾ cres 5686 “ cima 5687 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 Fincfn 8986 0cc0 11156 ℕcn 12267 ℕ0cn0 12528 Basecbs 17248 ↾s cress 17275 Scalarcsca 17301 0gc0g 17485 Ringcrg 20231 SubRingcsubrg 20570 algSccascl 21873 mPoly cmpl 21927 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-subrng 20547 df-subrg 20571 df-ascl 21876 df-psr 21930 df-mpl 21932 | 
| This theorem is referenced by: subrgasclcl 22092 subrg1ascl 22263 | 
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