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Theorem rgspnval 41524
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 6849 . 2 (πœ‘ β†’ (π‘β€˜π΄) = ((RingSpanβ€˜π‘…)β€˜π΄))
4 rgspnval.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ Ring)
5 elex 3466 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ V)
6 fveq2 6847 . . . . . . . 8 (π‘Ž = 𝑅 β†’ (Baseβ€˜π‘Ž) = (Baseβ€˜π‘…))
76pweqd 4582 . . . . . . 7 (π‘Ž = 𝑅 β†’ 𝒫 (Baseβ€˜π‘Ž) = 𝒫 (Baseβ€˜π‘…))
8 fveq2 6847 . . . . . . . . 9 (π‘Ž = 𝑅 β†’ (SubRingβ€˜π‘Ž) = (SubRingβ€˜π‘…))
9 rabeq 3424 . . . . . . . . 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 4917 . . . . . . 7 (π‘Ž = 𝑅 β†’ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑} = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
127, 11mpteq12dv 5201 . . . . . 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 20237 . . . . . 6 RingSpan = (π‘Ž ∈ V ↦ (𝑏 ∈ 𝒫 (Baseβ€˜π‘Ž) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘Ž) ∣ 𝑏 βŠ† 𝑑}))
14 fvex 6860 . . . . . . . 8 (Baseβ€˜π‘…) ∈ V
1514pwex 5340 . . . . . . 7 𝒫 (Baseβ€˜π‘…) ∈ V
1615mptex 7178 . . . . . 6 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 6953 . . . . 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 6849 . . 3 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄))
20 eqid 2737 . . . 4 (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑}) = (𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})
21 sseq1 3974 . . . . . 6 (𝑏 = 𝐴 β†’ (𝑏 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝑑))
2221rabbidv 3418 . . . . 5 (𝑏 = 𝐴 β†’ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑} = {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
2322inteqd 4917 . . . 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 3989 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
2714elpw2 5307 . . . . 5 (𝐴 ∈ 𝒫 (Baseβ€˜π‘…) ↔ 𝐴 βŠ† (Baseβ€˜π‘…))
2826, 27sylibr 233 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝒫 (Baseβ€˜π‘…))
29 eqid 2737 . . . . . . . . 9 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3029subrgid 20240 . . . . . . . 8 (𝑅 ∈ Ring β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
314, 30syl 17 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…))
3225, 31eqeltrd 2838 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (SubRingβ€˜π‘…))
33 sseq2 3975 . . . . . . 7 (𝑑 = 𝐡 β†’ (𝐴 βŠ† 𝑑 ↔ 𝐴 βŠ† 𝐡))
3433rspcev 3584 . . . . . 6 ((𝐡 ∈ (SubRingβ€˜π‘…) ∧ 𝐴 βŠ† 𝐡) β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
3532, 24, 34syl2anc 585 . . . . 5 (πœ‘ β†’ βˆƒπ‘‘ ∈ (SubRingβ€˜π‘…)𝐴 βŠ† 𝑑)
36 intexrab 5302 . . . . 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 6976 . . 3 (πœ‘ β†’ ((𝑏 ∈ 𝒫 (Baseβ€˜π‘…) ↦ ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝑏 βŠ† 𝑑})β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
3919, 38eqtrd 2777 . 2 (πœ‘ β†’ ((RingSpanβ€˜π‘…)β€˜π΄) = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
401, 3, 393eqtrd 2781 1 (πœ‘ β†’ π‘ˆ = ∩ {𝑑 ∈ (SubRingβ€˜π‘…) ∣ 𝐴 βŠ† 𝑑})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βŠ† wss 3915  π’« cpw 4565  βˆ© cint 4912   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  Ringcrg 19971  SubRingcsubrg 20234  RingSpancrgspn 20235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mgp 19904  df-ur 19921  df-ring 19973  df-subrg 20236  df-rgspn 20237
This theorem is referenced by:  rgspncl  41525  rgspnssid  41526  rgspnmin  41527
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