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Theorem rgspnval 41481
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 6844 . 2 (𝜑 → (𝑁𝐴) = ((RingSpan‘𝑅)‘𝐴))
4 rgspnval.r . . . . 5 (𝜑𝑅 ∈ Ring)
5 elex 3463 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
6 fveq2 6842 . . . . . . . 8 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
76pweqd 4577 . . . . . . 7 (𝑎 = 𝑅 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑅))
8 fveq2 6842 . . . . . . . . 9 (𝑎 = 𝑅 → (SubRing‘𝑎) = (SubRing‘𝑅))
9 rabeq 3421 . . . . . . . . 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 4912 . . . . . . 7 (𝑎 = 𝑅 {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡})
127, 11mpteq12dv 5196 . . . . . 6 (𝑎 = 𝑅 → (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}) = (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}))
13 df-rgspn 20221 . . . . . 6 RingSpan = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 (Base‘𝑎) ↦ {𝑡 ∈ (SubRing‘𝑎) ∣ 𝑏𝑡}))
14 fvex 6855 . . . . . . . 8 (Base‘𝑅) ∈ V
1514pwex 5335 . . . . . . 7 𝒫 (Base‘𝑅) ∈ V
1615mptex 7173 . . . . . 6 (𝑏 ∈ 𝒫 (Base‘𝑅) ↦ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡}) ∈ V
1712, 13, 16fvmpt 6948 . . . . 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 6844 . . 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 3969 . . . . . 6 (𝑏 = 𝐴 → (𝑏𝑡𝐴𝑡))
2221rabbidv 3415 . . . . 5 (𝑏 = 𝐴 → {𝑡 ∈ (SubRing‘𝑅) ∣ 𝑏𝑡} = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})
2322inteqd 4912 . . . 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 3984 . . . . 5 (𝜑𝐴 ⊆ (Base‘𝑅))
2714elpw2 5302 . . . . 5 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2826, 27sylibr 233 . . . 4 (𝜑𝐴 ∈ 𝒫 (Base‘𝑅))
29 eqid 2736 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
3029subrgid 20224 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
314, 30syl 17 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
3225, 31eqeltrd 2838 . . . . . 6 (𝜑𝐵 ∈ (SubRing‘𝑅))
33 sseq2 3970 . . . . . . 7 (𝑡 = 𝐵 → (𝐴𝑡𝐴𝐵))
3433rspcev 3581 . . . . . 6 ((𝐵 ∈ (SubRing‘𝑅) ∧ 𝐴𝐵) → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
3532, 24, 34syl2anc 584 . . . . 5 (𝜑 → ∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡)
36 intexrab 5297 . . . . 5 (∃𝑡 ∈ (SubRing‘𝑅)𝐴𝑡 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3735, 36sylib 217 . . . 4 (𝜑 {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡} ∈ V)
3820, 23, 28, 37fvmptd3 6971 . . 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 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560   cint 4907  cmpt 5188  cfv 6496  Basecbs 17083  Ringcrg 19964  SubRingcsubrg 20218  RingSpancrgspn 20219
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-rgspn 20221
This theorem is referenced by:  rgspncl  41482  rgspnssid  41483  rgspnmin  41484
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