Theorem rgspnval 39109

Description: Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)

Hypotheses

Ref	Expression
rgspnval.r	⊢ (𝜑 → 𝑅 ∈ Ring)
rgspnval.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
rgspnval.ss	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
rgspnval.n	⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))
rgspnval.sp	⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))

Assertion

Ref	Expression
rgspnval	⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})

Distinct variable groups:   𝜑,𝑡   𝑡,𝑅   𝑡,𝐵   𝑡,𝐴

Allowed substitution hints:   𝑈(𝑡)   𝑁(𝑡)

Proof of Theorem rgspnval

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rgspnval.sp	. 2 ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))
2		rgspnval.n	. . 3 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))
3	2	fveq1d 6495	. 2 ⊢ (𝜑 → (𝑁‘𝐴) = ((RingSpan‘𝑅)‘𝐴))
4		rgspnval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		elex 3427	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
6		fveq2 6493	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7	6	pweqd 4421	. . . . . . 7 ⊢ (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8		fveq2 6493	. . . . . . . . 9 ⊢ (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9		rabeq 3400	. . . . . . . . 9 ⊢ ((SubRing‘𝑎) = (SubRing‘𝑅) → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
11	10	inteqd 4748	. . . . . . 7 ⊢ (𝑎 = 𝑅 → ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
12	7, 11	mpteq12dv 5006	. . . . . 6 ⊢ (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
13		df-rgspn 19247	. . . . . 6 ⊢ RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}))
14		fvex 6506	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
15	14	pwex 5128	. . . . . . 7 ⊢ 𝒫 (Base‘𝑅) ∈ V
16	15	mptex 6806	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 6589	. . . . 5 ⊢ (𝑅 ∈ V → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
18	4, 5, 17	3syl 18	. . . 4 ⊢ (𝜑 → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
19	18	fveq1d 6495	. . 3 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴))
20		eqid 2772	. . . 4 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
21		sseq1 3878	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡))
22	21	rabbidv 3397	. . . . 5 ⊢ (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
23	22	inteqd 4748	. . . 4 ⊢ (𝑏 = 𝐴 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
24		rgspnval.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
25		rgspnval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
26	24, 25	sseqtrd 3893	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝑅))
27	14	elpw2 5098	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
28	26, 27	sylibr 226	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (Base‘𝑅))
29		eqid 2772	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	29	subrgid 19250	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
31	4, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
32	25, 31	eqeltrd 2860	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
33		sseq2 3879	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵))
34	33	rspcev 3529	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
35	32, 24, 34	syl2anc 576	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
36		intexrab 5093	. . . . 5 ⊢ (∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
37	35, 36	sylib 210	. . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
38	20, 23, 28, 37	fvmptd3 6611	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
39	19, 38	eqtrd 2808	. 2 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
40	1, 3, 39	3eqtrd 2812	1 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2048  ∃wrex 3083  {crab 3086  Vcvv 3409   ⊆ wss 3825  𝒫 cpw 4416  ∩ cint 4743   ↦ cmpt 5002  ‘cfv 6182  Basecbs 16329  Ringcrg 19010  SubRingcsubrg 19244  RingSpancrgspn 19245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-0g 16561  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mgp 18953  df-ur 18965  df-ring 19012  df-subrg 19246  df-rgspn 19247

This theorem is referenced by:  rgspncl  39110  rgspnssid  39111  rgspnmin  39112