Theorem rgspnval 41256

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 6827	. 2 ⊢ (𝜑 → (𝑁‘𝐴) = ((RingSpan‘𝑅)‘𝐴))
4		rgspnval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		elex 3459	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
6		fveq2 6825	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7	6	pweqd 4564	. . . . . . 7 ⊢ (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8		fveq2 6825	. . . . . . . . 9 ⊢ (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9		rabeq 3417	. . . . . . . . 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 4899	. . . . . . 7 ⊢ (𝑎 = 𝑅 → ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
12	7, 11	mpteq12dv 5183	. . . . . 6 ⊢ (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
13		df-rgspn 20128	. . . . . 6 ⊢ RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}))
14		fvex 6838	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
15	14	pwex 5323	. . . . . . 7 ⊢ 𝒫 (Base‘𝑅) ∈ V
16	15	mptex 7155	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 6931	. . . . 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 6827	. . 3 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴))
20		eqid 2736	. . . 4 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
21		sseq1 3957	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡))
22	21	rabbidv 3411	. . . . 5 ⊢ (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
23	22	inteqd 4899	. . . 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 3972	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝑅))
27	14	elpw2 5289	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
28	26, 27	sylibr 233	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (Base‘𝑅))
29		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	29	subrgid 20131	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
31	4, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
32	25, 31	eqeltrd 2837	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
33		sseq2 3958	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵))
34	33	rspcev 3570	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
35	32, 24, 34	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
36		intexrab 5284	. . . . 5 ⊢ (∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
37	35, 36	sylib 217	. . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
38	20, 23, 28, 37	fvmptd3 6954	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
39	19, 38	eqtrd 2776	. 2 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
40	1, 3, 39	3eqtrd 2780	1 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  {crab 3403  Vcvv 3441   ⊆ wss 3898  𝒫 cpw 4547  ∩ cint 4894   ↦ cmpt 5175  ‘cfv 6479  Basecbs 17009  Ringcrg 19878  SubRingcsubrg 20125  RingSpancrgspn 20126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-mgp 19816  df-ur 19833  df-ring 19880  df-subrg 20127  df-rgspn 20128

This theorem is referenced by:  rgspncl  41257  rgspnssid  41258  rgspnmin  41259