MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rgspnval Structured version   Visualization version   GIF version

Theorem rgspnval 20613
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 (𝜑𝑅 ∈ Ring)
rgspnval.b (𝜑𝐵 = (Base‘𝑅))
rgspnval.ss (𝜑𝐴𝐵)
rgspnval.n (𝜑𝑁 = (RingSpan‘𝑅))
rgspnval.sp (𝜑𝑈 = (𝑁𝐴))
Assertion
Ref Expression
rgspnval (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
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.
StepHypRef Expression
1 rgspnval.sp . 2 (𝜑𝑈 = (𝑁𝐴))
2 rgspnval.n . . 3 (𝜑𝑁 = (RingSpan‘𝑅))
32fveq1d 6907 . 2 (𝜑 → (𝑁𝐴) = ((RingSpan‘𝑅)‘𝐴))
4 rgspnval.r . . . . 5 (𝜑𝑅 ∈ Ring)
5 elex 3500 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
6 fveq2 6905 . . . . . . . 8 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
76pweqd 4616 . . . . . . 7 (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8 fveq2 6905 . . . . . . . . 9 (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9 rabeq 3450 . . . . . . . . 9 ((SubRing‘𝑎) = (SubRing‘𝑅) → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
108, 9syl 17 . . . . . . . 8 (𝑎 = 𝑅 → {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
1110inteqd 4950 . . . . . . 7 (𝑎 = 𝑅 {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
127, 11mpteq12dv 5232 . . . . . 6 (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
13 df-rgspn 20612 . . . . . 6 RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}))
14 fvex 6918 . . . . . . . 8 (Base‘𝑅) ∈ V
1514pwex 5379 . . . . . . 7 𝒫 (Base‘𝑅) ∈ V
1615mptex 7244 . . . . . 6 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) ∈ V
1712, 13, 16fvmpt 7015 . . . . 5 (𝑅 ∈ V → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
184, 5, 173syl 18 . . . 4 (𝜑 → (RingSpan‘𝑅) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
1918fveq1d 6907 . . 3 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴))
20 eqid 2736 . . . 4 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
21 sseq1 4008 . . . . . 6 (𝑏 = 𝐴 → (𝑏𝑡𝐴𝑡))
2221rabbidv 3443 . . . . 5 (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
2322inteqd 4950 . . . 4 (𝑏 = 𝐴 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
24 rgspnval.ss . . . . . 6 (𝜑𝐴𝐵)
25 rgspnval.b . . . . . 6 (𝜑𝐵 = (Base‘𝑅))
2624, 25sseqtrd 4019 . . . . 5 (𝜑𝐴 ⊆ (Base‘𝑅))
2714elpw2 5333 . . . . 5 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2826, 27sylibr 234 . . . 4 (𝜑𝐴 ∈ 𝒫 (Base‘𝑅))
29 eqid 2736 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
3029subrgid 20574 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
314, 30syl 17 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
3225, 31eqeltrd 2840 . . . . . 6 (𝜑𝐵 ∈ (SubRing‘𝑅))
33 sseq2 4009 . . . . . . 7 (𝑡 = 𝐵 → (𝐴𝑡𝐴𝐵))
3433rspcev 3621 . . . . . 6 ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
3532, 24, 34syl2anc 584 . . . . 5 (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
36 intexrab 5346 . . . . 5 (∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3735, 36sylib 218 . . . 4 (𝜑 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3820, 23, 28, 37fvmptd3 7038 . . 3 (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
3919, 38eqtrd 2776 . 2 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
401, 3, 393eqtrd 2780 1 (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   cint 4945  cmpt 5224  cfv 6560  Basecbs 17248  Ringcrg 20231  SubRingcsubrg 20570  RingSpancrgspn 20611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mgp 20139  df-ur 20180  df-ring 20233  df-subrg 20571  df-rgspn 20612
This theorem is referenced by:  rgspncl  20614  rgspnssid  20615  rgspnmin  20616  elrgspnlem4  33250
  Copyright terms: Public domain W3C validator