Theorem rgspnval 20629

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 6909	. 2 ⊢ (𝜑 → (𝑁‘𝐴) = ((RingSpan‘𝑅)‘𝐴))
4		rgspnval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		elex 3499	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
6		fveq2 6907	. . . . . . . 8 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7	6	pweqd 4622	. . . . . . 7 ⊢ (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8		fveq2 6907	. . . . . . . . 9 ⊢ (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9		rabeq 3448	. . . . . . . . 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 4956	. . . . . . 7 ⊢ (𝑎 = 𝑅 → ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡} = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
12	7, 11	mpteq12dv 5239	. . . . . 6 ⊢ (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}))
13		df-rgspn 20628	. . . . . 6 ⊢ RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ ∩ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏 ⊆ 𝑡}))
14		fvex 6920	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
15	14	pwex 5386	. . . . . . 7 ⊢ 𝒫 (Base‘𝑅) ∈ V
16	15	mptex 7243	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 7016	. . . . 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 6909	. . 3 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴))
20		eqid 2735	. . . 4 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})
21		sseq1 4021	. . . . . 6 ⊢ (𝑏 = 𝐴 → (𝑏 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑡))
22	21	rabbidv 3441	. . . . 5 ⊢ (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
23	22	inteqd 4956	. . . 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 4036	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝑅))
27	14	elpw2 5340	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
28	26, 27	sylibr 234	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (Base‘𝑅))
29		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	29	subrgid 20590	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
31	4, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
32	25, 31	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
33		sseq2 4022	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝐵))
34	33	rspcev 3622	. . . . . 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 5353	. . . . 5 ⊢ (∃𝑡 ∈ (SubRing‘𝑅)𝐴 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
37	35, 36	sylib 218	. . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡} ∈ V)
38	20, 23, 28, 37	fvmptd3 7039	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏 ⊆ 𝑡})‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
39	19, 38	eqtrd 2775	. 2 ⊢ (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
40	1, 3, 39	3eqtrd 2779	1 ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∩ cint 4951   ↦ cmpt 5231  ‘cfv 6563  Basecbs 17245  Ringcrg 20251  SubRingcsubrg 20586  RingSpancrgspn 20627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mgp 20153  df-ur 20200  df-ring 20253  df-subrg 20587  df-rgspn 20628

This theorem is referenced by:  rgspncl  20630  rgspnssid  20631  rgspnmin  20632  elrgspnlem4  33235