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Theorem rgspnval 38257
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 6334 . 2 (𝜑 → (𝑁𝐴) = ((RingSpan‘𝑅)‘𝐴))
4 rgspnval.r . . . . 5 (𝜑𝑅 ∈ Ring)
5 elex 3361 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
6 fveq2 6332 . . . . . . . 8 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
76pweqd 4300 . . . . . . 7 (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8 fveq2 6332 . . . . . . . . 9 (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9 rabeq 3341 . . . . . . . . 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 4614 . . . . . . 7 (𝑎 = 𝑅 {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
127, 11mpteq12dv 4865 . . . . . 6 (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
13 df-rgspn 18988 . . . . . 6 RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}))
14 fvex 6342 . . . . . . . 8 (Base‘𝑅) ∈ V
1514pwex 4976 . . . . . . 7 𝒫 (Base‘𝑅) ∈ V
1615mptex 6629 . . . . . 6 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) ∈ V
1712, 13, 16fvmpt 6424 . . . . 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 6334 . . 3 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴))
20 rgspnval.ss . . . . . 6 (𝜑𝐴𝐵)
21 rgspnval.b . . . . . 6 (𝜑𝐵 = (Base‘𝑅))
2220, 21sseqtrd 3788 . . . . 5 (𝜑𝐴 ⊆ (Base‘𝑅))
2314elpw2 4956 . . . . 5 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2422, 23sylibr 224 . . . 4 (𝜑𝐴 ∈ 𝒫 (Base‘𝑅))
25 eqid 2770 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
2625subrgid 18991 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
274, 26syl 17 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
2821, 27eqeltrd 2849 . . . . . 6 (𝜑𝐵 ∈ (SubRing‘𝑅))
29 sseq2 3774 . . . . . . 7 (𝑡 = 𝐵 → (𝐴𝑡𝐴𝐵))
3029rspcev 3458 . . . . . 6 ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
3128, 20, 30syl2anc 565 . . . . 5 (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
32 intexrab 4951 . . . . 5 (∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3331, 32sylib 208 . . . 4 (𝜑 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
34 sseq1 3773 . . . . . . 7 (𝑏 = 𝐴 → (𝑏𝑡𝐴𝑡))
3534rabbidv 3338 . . . . . 6 (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
3635inteqd 4614 . . . . 5 (𝑏 = 𝐴 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
37 eqid 2770 . . . . 5 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
3836, 37fvmptg 6422 . . . 4 ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V) → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
3924, 33, 38syl2anc 565 . . 3 (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
4019, 39eqtrd 2804 . 2 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
411, 3, 403eqtrd 2808 1 (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  𝒫 cpw 4295   cint 4609  cmpt 4861  cfv 6031  Basecbs 16063  Ringcrg 18754  SubRingcsubrg 18985  RingSpancrgspn 18986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mgp 18697  df-ur 18709  df-ring 18756  df-subrg 18987  df-rgspn 18988
This theorem is referenced by:  rgspncl  38258  rgspnssid  38259  rgspnmin  38260
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