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Theorem rgspnval 42864
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 6893 . 2 (𝜑 → (𝑁𝐴) = ((RingSpan‘𝑅)‘𝐴))
4 rgspnval.r . . . . 5 (𝜑𝑅 ∈ Ring)
5 elex 3483 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
6 fveq2 6891 . . . . . . . 8 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
76pweqd 4615 . . . . . . 7 (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8 fveq2 6891 . . . . . . . . 9 (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9 rabeq 3435 . . . . . . . . 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 4952 . . . . . . 7 (𝑎 = 𝑅 {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
127, 11mpteq12dv 5235 . . . . . 6 (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
13 df-rgspn 20548 . . . . . 6 RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}))
14 fvex 6904 . . . . . . . 8 (Base‘𝑅) ∈ V
1514pwex 5375 . . . . . . 7 𝒫 (Base‘𝑅) ∈ V
1615mptex 7230 . . . . . 6 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) ∈ V
1712, 13, 16fvmpt 6999 . . . . 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 6893 . . 3 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴))
20 eqid 2726 . . . 4 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
21 sseq1 4005 . . . . . 6 (𝑏 = 𝐴 → (𝑏𝑡𝐴𝑡))
2221rabbidv 3428 . . . . 5 (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
2322inteqd 4952 . . . 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 4020 . . . . 5 (𝜑𝐴 ⊆ (Base‘𝑅))
2714elpw2 5343 . . . . 5 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2826, 27sylibr 233 . . . 4 (𝜑𝐴 ∈ 𝒫 (Base‘𝑅))
29 eqid 2726 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
3029subrgid 20551 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
314, 30syl 17 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
3225, 31eqeltrd 2826 . . . . . 6 (𝜑𝐵 ∈ (SubRing‘𝑅))
33 sseq2 4006 . . . . . . 7 (𝑡 = 𝐵 → (𝐴𝑡𝐴𝐵))
3433rspcev 3608 . . . . . 6 ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
3532, 24, 34syl2anc 582 . . . . 5 (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
36 intexrab 5338 . . . . 5 (∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3735, 36sylib 217 . . . 4 (𝜑 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3820, 23, 28, 37fvmptd3 7022 . . 3 (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
3919, 38eqtrd 2766 . 2 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
401, 3, 393eqtrd 2770 1 (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wrex 3060  {crab 3420  Vcvv 3463  wss 3947  𝒫 cpw 4598   cint 4947  cmpt 5227  cfv 6544  Basecbs 17206  Ringcrg 20210  SubRingcsubrg 20545  RingSpancrgspn 20546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-nn 12257  df-2 12319  df-sets 17159  df-slot 17177  df-ndx 17189  df-base 17207  df-ress 17236  df-plusg 17272  df-0g 17449  df-mgm 18626  df-sgrp 18705  df-mnd 18721  df-mgp 20112  df-ur 20159  df-ring 20212  df-subrg 20547  df-rgspn 20548
This theorem is referenced by:  rgspncl  42865  rgspnssid  42866  rgspnmin  42867
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