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Theorem rgspnval 42488
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 6887 . 2 (πœ‘ β†’ (π‘β€˜π΄) = ((RingSpanβ€˜π‘…)β€˜π΄))
4 rgspnval.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ Ring)
5 elex 3487 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ V)
6 fveq2 6885 . . . . . . . 8 (π‘Ž = 𝑅 β†’ (Baseβ€˜π‘Ž) = (Baseβ€˜π‘…))
76pweqd 4614 . . . . . . 7 (π‘Ž = 𝑅 β†’ 𝒫 (Baseβ€˜π‘Ž) = 𝒫 (Baseβ€˜π‘…))
8 fveq2 6885 . . . . . . . . 9 (π‘Ž = 𝑅 β†’ (SubRingβ€˜π‘Ž) = (SubRingβ€˜π‘…))
9 rabeq 3440 . . . . . . . . 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 4948 . . . . . . 7 (π‘Ž = 𝑅 β†’ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑} = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
127, 11mpteq12dv 5232 . . . . . 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 20472 . . . . . 6 RingSpan = (π‘Ž ∈ V ↦ (𝑏 ∈ 𝒫 (Baseβ€˜π‘Ž) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑}))
14 fvex 6898 . . . . . . . 8 (Baseβ€˜π‘…) ∈ V
1514pwex 5371 . . . . . . 7 𝒫 (Baseβ€˜π‘…) ∈ V
1615mptex 7220 . . . . . 6 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 6992 . . . . 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 6887 . . 3 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄))
20 eqid 2726 . . . 4 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
21 sseq1 4002 . . . . . 6 (𝑏 = 𝐴 β†’ (𝑏 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝑑))
2221rabbidv 3434 . . . . 5 (𝑏 = 𝐴 β†’ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑} = {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
2322inteqd 4948 . . . 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 4017 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2714elpw2 5338 . . . . 5 (𝐴 ∈ 𝒫 (Baseβ€˜π‘…) ↔ 𝐴 βŠ† (Baseβ€˜π‘…))
2826, 27sylibr 233 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝒫 (Baseβ€˜π‘…))
29 eqid 2726 . . . . . . . . 9 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3029subrgid 20475 . . . . . . . 8 (𝑅 ∈ Ring β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
314, 30syl 17 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
3225, 31eqeltrd 2827 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
33 sseq2 4003 . . . . . . 7 (𝑑 = 𝐡 β†’ (𝐴 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝐡))
3433rspcev 3606 . . . . . 6 ((𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐴 βŠ† 𝐡) β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
3532, 24, 34syl2anc 583 . . . . 5 (πœ‘ β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
36 intexrab 5333 . . . . 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 7015 . . 3 (πœ‘ β†’ ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
3919, 38eqtrd 2766 . 2 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
401, 3, 393eqtrd 2770 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 3064  {crab 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597  βˆ© cint 4943   ↦ cmpt 5224  β€˜cfv 6537  Basecbs 17153  Ringcrg 20138  SubRingcsubrg 20469  RingSpancrgspn 20470
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mgp 20040  df-ur 20087  df-ring 20140  df-subrg 20471  df-rgspn 20472
This theorem is referenced by:  rgspncl  42489  rgspnssid  42490  rgspnmin  42491
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