subrgascl - Metamath Proof Explorer

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Theorem subrgascl 22029

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 21846	. . 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 20546	. . . . . 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 7444	. . . . . . . 8 ⊢ 𝐻 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V)
13	9, 10, 12	mplsca 21978	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈))
14	13	fveq2d 6885	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈)))
15	8, 14	eqtrd 2771	. . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈)))
16	15	fneq2d 6637	. . 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 21846	. . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃))
22		subrgascl.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
23		subrgrcl 20541	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
24	5, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
25	22, 10, 24	mplsca 21978	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
26	25	fveq2d 6885	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
27	26	fneq2d 6637	. . . 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 20537	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
31	5, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
32		fnssres 6666	. . 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 6900	. . . 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 20544	. . . . . . . 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 4526	. . . . . 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 5220	. . . 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 3963	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅))
46	22, 42, 36, 29, 18, 43, 44, 45	mplascl 22027	. . . 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 20539	. . . . . . 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 2821	. . . . . 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 22027	. . . 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 2781	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥))
56	35, 55	eqtr2d 2772	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥))
57	17, 33, 56	eqfnfvd 7029	1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3420 Vcvv 3464 ⊆ wss 3931 ifcif 4505 {csn 4606 ↦ cmpt 5206 × cxp 5657 ^◡ccnv 5658 ↾ cres 5661 “ cima 5662 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 Fincfn 8964 0cc0 11134 ℕcn 12245 ℕ₀cn0 12506 Basecbs 17233 ↾_s cress 17256 Scalarcsca 17279 0_gc0g 17458 Ringcrg 20198 SubRingcsubrg 20534 algSccascl 21817 mPoly cmpl 21871

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-subrng 20511 df-subrg 20535 df-ascl 21820 df-psr 21874 df-mpl 21876

This theorem is referenced by: subrgasclcl 22030 subrg1ascl 22201

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