subrgascl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > subrgascl		Structured version Visualization version GIF version

Theorem subrgascl 19498

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
subrgascl	⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))

Proof of Theorem subrgascl Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgascl.c	. . . 4 ⊢ 𝐶 = (algSc‘𝑈)
2		eqid 2622	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
3		eqid 2622	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
4	1, 2, 3	asclfn 19336	. . 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 18789	. . . . . 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 6678	. . . . . . . . 9 ⊢ (𝑅 ↾_s 𝑇) ∈ V
12	6, 11	eqeltri 2697	. . . . . . . 8 ⊢ 𝐻 ∈ V
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V)
14	9, 10, 13	mplsca 19445	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈))
15	14	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈)))
16	8, 15	eqtrd 2656	. . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈)))
17	16	fneq2d 5982	. . 3 ⊢ (𝜑 → (𝐶 Fn 𝑇 ↔ 𝐶 Fn (Base‘(Scalar‘𝑈))))
18	4, 17	mpbiri 248	. 2 ⊢ (𝜑 → 𝐶 Fn 𝑇)
19		subrgascl.a	. . . . 5 ⊢ 𝐴 = (algSc‘𝑃)
20		eqid 2622	. . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
21		eqid 2622	. . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
22	19, 20, 21	asclfn 19336	. . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃))
23		subrgascl.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
24		subrgrcl 18785	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
25	5, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
26	23, 10, 25	mplsca 19445	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
27	26	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
28	27	fneq2d 5982	. . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃))))
29	22, 28	mpbiri 248	. . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅))
30		eqid 2622	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
31	30	subrgss 18781	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
32	5, 31	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
33		fnssres 6004	. . 3 ⊢ ((𝐴 Fn (Base‘𝑅) ∧ 𝑇 ⊆ (Base‘𝑅)) → (𝐴 ↾ 𝑇) Fn 𝑇)
34	29, 32, 33	syl2anc 693	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝑇) Fn 𝑇)
35		fvres 6207	. . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥))
36	35	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥))
37		eqid 2622	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
38	6, 37	subrg0 18787	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0_g‘𝑅) = (0_g‘𝐻))
39	5, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘𝐻))
40	39	ifeq2d 4105	. . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻)))
41	40	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻)))
42	41	mpteq2dv 4745	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅))) = (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻))))
43		eqid 2622	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
44	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐼 ∈ 𝑊)
45	25	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑅 ∈ Ring)
46	32	sselda 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅))
47	23, 43, 37, 30, 19, 44, 45, 46	mplascl 19496	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅))))
48		eqid 2622	. . . . 5 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
49		eqid 2622	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
50	6	subrgring 18783	. . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
51	5, 50	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring)
52	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐻 ∈ Ring)
53	8	eleq2d 2687	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ (Base‘𝐻)))
54	53	biimpa 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝐻))
55	9, 43, 48, 49, 1, 44, 52, 54	mplascl 19496	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻))))
56	42, 47, 55	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥))
57	36, 56	eqtr2d 2657	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥))
58	18, 34, 57	eqfnfvd 6314	1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 ifcif 4086 {csn 4177 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 ↾ cres 5116 “ cima 5117 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 Fincfn 7955 0cc0 9936 ℕcn 11020 ℕ₀cn0 11292 Basecbs 15857 ↾_s cress 15858 Scalarcsca 15944 0_gc0g 16100 Ringcrg 18547 SubRingcsubrg 18776 algSccascl 19311 mPoly cmpl 19353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-ascl 19314 df-psr 19356 df-mpl 19358

This theorem is referenced by: subrgasclcl 19499 subrg1ascl 19629

W3C validator