subrgascl - Metamath Proof Explorer

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Theorem subrgascl 22113

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 2740	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
3		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
4	1, 2, 3	asclfn 21924	. . 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 20609	. . . . . 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 7482	. . . . . . . 8 ⊢ 𝐻 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V)
13	9, 10, 12	mplsca 22056	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈))
14	13	fveq2d 6924	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈)))
15	8, 14	eqtrd 2780	. . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈)))
16	15	fneq2d 6673	. . 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 2740	. . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
20		eqid 2740	. . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
21	18, 19, 20	asclfn 21924	. . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃))
22		subrgascl.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
23		subrgrcl 20604	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
24	5, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
25	22, 10, 24	mplsca 22056	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
26	25	fveq2d 6924	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
27	26	fneq2d 6673	. . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃))))
28	21, 27	mpbiri 258	. . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅))
29		eqid 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	29	subrgss 20600	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
31	5, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
32		fnssres 6703	. . 3 ⊢ ((𝐴 Fn (Base‘𝑅) ∧ 𝑇 ⊆ (Base‘𝑅)) → (𝐴 ↾ 𝑇) Fn 𝑇)
33	28, 31, 32	syl2anc 583	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝑇) Fn 𝑇)
34		fvres 6939	. . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥))
35	34	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥))
36		eqid 2740	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
37	6, 36	subrg0 20607	. . . . . . . 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 4568	. . . . . 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 5268	. . . 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 2740	. . . . 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 4008	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅))
46	22, 42, 36, 29, 18, 43, 44, 45	mplascl 22111	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0_g‘𝑅))))
47		eqid 2740	. . . . 5 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
48		eqid 2740	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
49	6	subrgring 20602	. . . . . . 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 2830	. . . . . 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 22111	. . . 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 2790	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥))
56	35, 55	eqtr2d 2781	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥))
57	17, 33, 56	eqfnfvd 7067	1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 Vcvv 3488 ⊆ wss 3976 ifcif 4548 {csn 4648 ↦ cmpt 5249 × cxp 5698 ^◡ccnv 5699 ↾ cres 5702 “ cima 5703 Fn wfn 6568 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 0cc0 11184 ℕcn 12293 ℕ₀cn0 12553 Basecbs 17258 ↾_s cress 17287 Scalarcsca 17314 0_gc0g 17499 Ringcrg 20260 SubRingcsubrg 20595 algSccascl 21895 mPoly cmpl 21949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-subrng 20572 df-subrg 20597 df-ascl 21898 df-psr 21952 df-mpl 21954

This theorem is referenced by: subrgasclcl 22114 subrg1ascl 22283

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