subrgascl - Metamath Proof Explorer

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Theorem subrgascl 22091

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 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 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
37	6, 36	subrg0 20580	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0_g‘𝑅) = (0_g‘𝐻))
38	5, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘𝐻))
39	38	ifeq2d 4545	. . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻)))
40	39	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝐻)))
41	40	mpteq2dv 5243	. . . 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))))
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, (0_g‘𝑅))))
47		eqid 2736	. . . . 5 ⊢ (0_g‘𝐻) = (0_g‘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, (0_g‘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 0_gc0g 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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