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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 2797 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
3 | eqid 2797 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
4 | 1, 2, 3 | asclfn 19802 | . . 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 19238 | . . . . . 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 7056 | . . . . . . . 8 ⊢ 𝐻 ∈ V |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V) |
13 | 9, 10, 12 | mplsca 19917 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈)) |
14 | 13 | fveq2d 6549 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈))) |
15 | 8, 14 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈))) |
16 | 15 | fneq2d 6324 | . . 3 ⊢ (𝜑 → (𝐶 Fn 𝑇 ↔ 𝐶 Fn (Base‘(Scalar‘𝑈)))) |
17 | 4, 16 | mpbiri 259 | . 2 ⊢ (𝜑 → 𝐶 Fn 𝑇) |
18 | subrgascl.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
19 | eqid 2797 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
20 | eqid 2797 | . . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
21 | 18, 19, 20 | asclfn 19802 | . . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃)) |
22 | subrgascl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
23 | subrgrcl 19234 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
24 | 5, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
25 | 22, 10, 24 | mplsca 19917 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
26 | 25 | fveq2d 6549 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
27 | 26 | fneq2d 6324 | . . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃)))) |
28 | 21, 27 | mpbiri 259 | . . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅)) |
29 | eqid 2797 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
30 | 29 | subrgss 19230 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
31 | 5, 30 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
32 | fnssres 6347 | . . 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 6564 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) | |
35 | 34 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) |
36 | eqid 2797 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
37 | 6, 36 | subrg0 19236 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
38 | 5, 37 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
39 | 38 | ifeq2d 4406 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
40 | 39 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
41 | 40 | mpteq2dv 5063 | . . . 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 2797 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
43 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐼 ∈ 𝑊) |
44 | 24 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑅 ∈ Ring) |
45 | 31 | sselda 3895 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅)) |
46 | 22, 42, 36, 29, 18, 43, 44, 45 | mplascl 19967 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) |
47 | eqid 2797 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
48 | eqid 2797 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
49 | 6 | subrgring 19232 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
50 | 5, 49 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
51 | 50 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐻 ∈ Ring) |
52 | 8 | eleq2d 2870 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ (Base‘𝐻))) |
53 | 52 | biimpa 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝐻)) |
54 | 9, 42, 47, 48, 1, 43, 51, 53 | mplascl 19967 | . . . 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 2843 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥)) |
56 | 35, 55 | eqtr2d 2834 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥)) |
57 | 17, 33, 56 | eqfnfvd 6677 | 1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 {crab 3111 Vcvv 3440 ⊆ wss 3865 ifcif 4387 {csn 4478 ↦ cmpt 5047 × cxp 5448 ◡ccnv 5449 ↾ cres 5452 “ cima 5453 Fn wfn 6227 ‘cfv 6232 (class class class)co 7023 ↑𝑚 cmap 8263 Fincfn 8364 0cc0 10390 ℕcn 11492 ℕ0cn0 11751 Basecbs 16316 ↾s cress 16317 Scalarcsca 16401 0gc0g 16546 Ringcrg 18991 SubRingcsubrg 19225 algSccascl 19777 mPoly cmpl 19825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-ofr 7275 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-seq 13224 df-hash 13545 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-tset 16417 df-0g 16548 df-gsum 16549 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mhm 17778 df-submnd 17779 df-grp 17868 df-minusg 17869 df-mulg 17986 df-subg 18034 df-ghm 18101 df-cntz 18192 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-subrg 19227 df-ascl 19780 df-psr 19828 df-mpl 19830 |
This theorem is referenced by: subrgasclcl 19970 subrg1ascl 20114 |
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