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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 2610 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
3 | eqid 2610 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
4 | 1, 2, 3 | asclfn 19157 | . . 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 18612 | . . . . . 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 | ovex 6577 | . . . . . . . . 9 ⊢ (𝑅 ↾s 𝑇) ∈ V | |
12 | 6, 11 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝐻 ∈ V |
13 | 12 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V) |
14 | 9, 10, 13 | mplsca 19266 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈)) |
15 | 14 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈))) |
16 | 8, 15 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈))) |
17 | 16 | fneq2d 5896 | . . 3 ⊢ (𝜑 → (𝐶 Fn 𝑇 ↔ 𝐶 Fn (Base‘(Scalar‘𝑈)))) |
18 | 4, 17 | mpbiri 247 | . 2 ⊢ (𝜑 → 𝐶 Fn 𝑇) |
19 | subrgascl.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
20 | eqid 2610 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
21 | eqid 2610 | . . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
22 | 19, 20, 21 | asclfn 19157 | . . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃)) |
23 | subrgascl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
24 | subrgrcl 18608 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
25 | 5, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
26 | 23, 10, 25 | mplsca 19266 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
27 | 26 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
28 | 27 | fneq2d 5896 | . . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃)))) |
29 | 22, 28 | mpbiri 247 | . . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅)) |
30 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
31 | 30 | subrgss 18604 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
32 | 5, 31 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
33 | fnssres 5918 | . . 3 ⊢ ((𝐴 Fn (Base‘𝑅) ∧ 𝑇 ⊆ (Base‘𝑅)) → (𝐴 ↾ 𝑇) Fn 𝑇) | |
34 | 29, 32, 33 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝑇) Fn 𝑇) |
35 | fvres 6117 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) | |
36 | 35 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) |
37 | eqid 2610 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
38 | 6, 37 | subrg0 18610 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
39 | 5, 38 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
40 | 39 | ifeq2d 4055 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
41 | 40 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
42 | 41 | mpteq2dv 4673 | . . . 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)))) |
43 | eqid 2610 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
44 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐼 ∈ 𝑊) |
45 | 25 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑅 ∈ Ring) |
46 | 32 | sselda 3568 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅)) |
47 | 23, 43, 37, 30, 19, 44, 45, 46 | mplascl 19317 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) |
48 | eqid 2610 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
49 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
50 | 6 | subrgring 18606 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
51 | 5, 50 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
52 | 51 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐻 ∈ Ring) |
53 | 8 | eleq2d 2673 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ (Base‘𝐻))) |
54 | 53 | biimpa 500 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝐻)) |
55 | 9, 43, 48, 49, 1, 44, 52, 54 | mplascl 19317 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻)))) |
56 | 42, 47, 55 | 3eqtr4d 2654 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥)) |
57 | 36, 56 | eqtr2d 2645 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥)) |
58 | 18, 34, 57 | eqfnfvd 6222 | 1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 ifcif 4036 {csn 4125 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 0cc0 9815 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 0gc0g 15923 Ringcrg 18370 SubRingcsubrg 18599 algSccascl 19132 mPoly cmpl 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-ascl 19135 df-psr 19177 df-mpl 19179 |
This theorem is referenced by: subrgasclcl 19320 subrg1ascl 19450 |
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