Theorem rgspnval 20698

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 6885	. 2 ⊢ (𝜑 → (𝑁‘𝐴) = ((RingSpan‘𝑅)‘𝐴))
4		rgspnval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		elex 3476	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
6		fveq2 6883	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7	6	pweqd 4580	. . . . . . 7 ⊢ (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8		fveq2 6883	. . . . . . . . 9 ⊢ (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9		rabeq 3430	. . . . . . . . 9 ⊢ ((SubRing‘𝑎) = (SubRing‘𝑅) → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
11	10	inteqd 4918	. . . . . . 7 ⊢ (𝑎 = 𝑅 → ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
12	7, 11	mpteq12dv 5199	. . . . . 6 ⊢ (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
13		df-rgspn 20697	. . . . . 6 ⊢ RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}))
14		fvex 6896	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
15	14	pwex 5353	. . . . . . 7 ⊢ 𝒫 (Base‘𝑅) ∈ V
16	15	mptex 7223	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 6991	. . . . 5 ⊢ (𝑅 ∈ V → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
18	4, 5, 17	3syl 19	. . . 4 ⊢ (𝜑 → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
19	18	fveq1d 6885	. . 3 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴))
20		eqid 2763	. . . 4 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
21		sseq1 3963	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡))
22	21	rabbidv 3423	. . . . 5 ⊢ (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
23	22	inteqd 4918	. . . 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 3974	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝑅))
27	14	elpw2 5306	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
28	26, 27	sylibr 237	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (Base‘𝑅))
29		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	29	subrgid 20659	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
31	4, 30	syl 18	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
32	25, 31	eqeltrd 2863	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
33		sseq2 3964	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵))
34	33	rspcev 3582	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴 ⊆ 𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
35	32, 24, 34	syl2anc 595	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡)
36		intexrab 5319	. . . . 5 ⊢ (∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
37	35, 36	sylib 221	. . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
38	20, 23, 28, 37	fvmptd3 7015	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
39	19, 38	eqtrd 2798	. 2 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
40	1, 3, 39	3eqtrd 2802	1 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3906  𝒫 cpw 4563  ∩ cint 4913   ↦ cmpt 5193  ‘cfv 6538  Basecbs 17270  Ringcrg 20316  SubRingcsubrg 20655  RingSpancrgspn 20696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mgp 20218  df-ur 20265  df-ring 20318  df-subrg 20656  df-rgspn 20697

This theorem is referenced by:  rgspncl  20699  rgspnssid  20700  rgspnmin  20701  elrgspnlem4  33546