Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rgspnval Structured version   Visualization version   GIF version

Theorem rgspnval 42641
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 (πœ‘ β†’ 𝑅 ∈ Ring)
rgspnval.b (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
rgspnval.ss (πœ‘ β†’ 𝐴 βŠ† 𝐡)
rgspnval.n (πœ‘ β†’ 𝑁 = (RingSpanβ€˜π‘…))
rgspnval.sp (πœ‘ β†’ π‘ˆ = (π‘β€˜π΄))
Assertion
Ref Expression
rgspnval (πœ‘ β†’ π‘ˆ = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
Distinct variable groups:   πœ‘,𝑑   𝑑,𝑅   𝑑,𝐡   𝑑,𝐴
Allowed substitution hints:   π‘ˆ(𝑑)   𝑁(𝑑)

Proof of Theorem rgspnval
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rgspnval.sp . 2 (πœ‘ β†’ π‘ˆ = (π‘β€˜π΄))
2 rgspnval.n . . 3 (πœ‘ β†’ 𝑁 = (RingSpanβ€˜π‘…))
32fveq1d 6904 . 2 (πœ‘ β†’ (π‘β€˜π΄) = ((RingSpanβ€˜π‘…)β€˜π΄))
4 rgspnval.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ Ring)
5 elex 3492 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ V)
6 fveq2 6902 . . . . . . . 8 (π‘Ž = 𝑅 β†’ (Baseβ€˜π‘Ž) = (Baseβ€˜π‘…))
76pweqd 4623 . . . . . . 7 (π‘Ž = 𝑅 β†’ 𝒫 (Baseβ€˜π‘Ž) = 𝒫 (Baseβ€˜π‘…))
8 fveq2 6902 . . . . . . . . 9 (π‘Ž = 𝑅 β†’ (SubRingβ€˜π‘Ž) = (SubRingβ€˜π‘…))
9 rabeq 3445 . . . . . . . . 9 ((SubRingβ€˜π‘Ž) = (SubRingβ€˜π‘…) β†’ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑} = {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
108, 9syl 17 . . . . . . . 8 (π‘Ž = 𝑅 β†’ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑} = {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
1110inteqd 4958 . . . . . . 7 (π‘Ž = 𝑅 β†’ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑} = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
127, 11mpteq12dv 5243 . . . . . 6 (π‘Ž = 𝑅 β†’ (𝑏 ∈ 𝒫 (Baseβ€˜π‘Ž) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑}) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}))
13 df-rgspn 20523 . . . . . 6 RingSpan = (π‘Ž ∈ V ↦ (𝑏 ∈ 𝒫 (Baseβ€˜π‘Ž) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑}))
14 fvex 6915 . . . . . . . 8 (Baseβ€˜π‘…) ∈ V
1514pwex 5384 . . . . . . 7 𝒫 (Baseβ€˜π‘…) ∈ V
1615mptex 7241 . . . . . 6 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 7010 . . . . 5 (𝑅 ∈ V β†’ (RingSpanβ€˜π‘…) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}))
184, 5, 173syl 18 . . . 4 (πœ‘ β†’ (RingSpanβ€˜π‘…) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}))
1918fveq1d 6904 . . 3 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄))
20 eqid 2728 . . . 4 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
21 sseq1 4007 . . . . . 6 (𝑏 = 𝐴 β†’ (𝑏 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝑑))
2221rabbidv 3438 . . . . 5 (𝑏 = 𝐴 β†’ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑} = {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
2322inteqd 4958 . . . 4 (𝑏 = 𝐴 β†’ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑} = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
24 rgspnval.ss . . . . . 6 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
25 rgspnval.b . . . . . 6 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
2624, 25sseqtrd 4022 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2714elpw2 5351 . . . . 5 (𝐴 ∈ 𝒫 (Baseβ€˜π‘…) ↔ 𝐴 βŠ† (Baseβ€˜π‘…))
2826, 27sylibr 233 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝒫 (Baseβ€˜π‘…))
29 eqid 2728 . . . . . . . . 9 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3029subrgid 20526 . . . . . . . 8 (𝑅 ∈ Ring β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
314, 30syl 17 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
3225, 31eqeltrd 2829 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
33 sseq2 4008 . . . . . . 7 (𝑑 = 𝐡 β†’ (𝐴 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝐡))
3433rspcev 3611 . . . . . 6 ((𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐴 βŠ† 𝐡) β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
3532, 24, 34syl2anc 582 . . . . 5 (πœ‘ β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
36 intexrab 5346 . . . . 5 (βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑 ↔ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑} ∈ V)
3735, 36sylib 217 . . . 4 (πœ‘ β†’ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑} ∈ V)
3820, 23, 28, 37fvmptd3 7033 . . 3 (πœ‘ β†’ ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
3919, 38eqtrd 2768 . 2 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
401, 3, 393eqtrd 2772 1 (πœ‘ β†’ π‘ˆ = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430  Vcvv 3473   βŠ† wss 3949  π’« cpw 4606  βˆ© cint 4953   ↦ cmpt 5235  β€˜cfv 6553  Basecbs 17189  Ringcrg 20187  SubRingcsubrg 20520  RingSpancrgspn 20521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-plusg 17255  df-0g 17432  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-mgp 20089  df-ur 20136  df-ring 20189  df-subrg 20522  df-rgspn 20523
This theorem is referenced by:  rgspncl  42642  rgspnssid  42643  rgspnmin  42644
  Copyright terms: Public domain W3C validator