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Theorem rgspnval 40105
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 6651 . 2 (𝜑 → (𝑁𝐴) = ((RingSpan‘𝑅)‘𝐴))
4 rgspnval.r . . . . 5 (𝜑𝑅 ∈ Ring)
5 elex 3462 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
6 fveq2 6649 . . . . . . . 8 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
76pweqd 4519 . . . . . . 7 (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8 fveq2 6649 . . . . . . . . 9 (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9 rabeq 3434 . . . . . . . . 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 4846 . . . . . . 7 (𝑎 = 𝑅 {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
127, 11mpteq12dv 5118 . . . . . 6 (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
13 df-rgspn 19531 . . . . . 6 RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}))
14 fvex 6662 . . . . . . . 8 (Base‘𝑅) ∈ V
1514pwex 5249 . . . . . . 7 𝒫 (Base‘𝑅) ∈ V
1615mptex 6967 . . . . . 6 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) ∈ V
1712, 13, 16fvmpt 6749 . . . . 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 6651 . . 3 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴))
20 eqid 2801 . . . 4 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
21 sseq1 3943 . . . . . 6 (𝑏 = 𝐴 → (𝑏𝑡𝐴𝑡))
2221rabbidv 3430 . . . . 5 (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
2322inteqd 4846 . . . 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 3958 . . . . 5 (𝜑𝐴 ⊆ (Base‘𝑅))
2714elpw2 5215 . . . . 5 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2826, 27sylibr 237 . . . 4 (𝜑𝐴 ∈ 𝒫 (Base‘𝑅))
29 eqid 2801 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
3029subrgid 19534 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
314, 30syl 17 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
3225, 31eqeltrd 2893 . . . . . 6 (𝜑𝐵 ∈ (SubRing‘𝑅))
33 sseq2 3944 . . . . . . 7 (𝑡 = 𝐵 → (𝐴𝑡𝐴𝐵))
3433rspcev 3574 . . . . . 6 ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
3532, 24, 34syl2anc 587 . . . . 5 (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
36 intexrab 5210 . . . . 5 (∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3735, 36sylib 221 . . . 4 (𝜑 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3820, 23, 28, 37fvmptd3 6772 . . 3 (𝜑 → ((𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
3919, 38eqtrd 2836 . 2 (𝜑 → ((RingSpan‘𝑅)‘𝐴) = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
401, 3, 393eqtrd 2840 1 (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  𝒫 cpw 4500   cint 4841  cmpt 5113  cfv 6328  Basecbs 16479  Ringcrg 19294  SubRingcsubrg 19528  RingSpancrgspn 19529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mgp 19237  df-ur 19249  df-ring 19296  df-subrg 19530  df-rgspn 19531
This theorem is referenced by:  rgspncl  40106  rgspnssid  40107  rgspnmin  40108
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