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Theorem rloc1r 33258
Description: The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
rloc0g.1 0 = (0g𝑅)
rloc0g.2 1 = (1r𝑅)
rloc0g.3 𝐿 = (𝑅 RLocal 𝑆)
rloc0g.4 = (𝑅 ~RL 𝑆)
rloc0g.5 (𝜑𝑅 ∈ CRing)
rloc0g.6 (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
rloc1r.i 𝐼 = [⟨ 1 , 1 ⟩]
Assertion
Ref Expression
rloc1r (𝜑𝐼 = (1r𝐿))

Proof of Theorem rloc1r
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rloc1r.i . 2 𝐼 = [⟨ 1 , 1 ⟩]
2 eqid 2734 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 eqid 2734 . . . . 5 (.r𝑅) = (.r𝑅)
4 eqid 2734 . . . . 5 (+g𝑅) = (+g𝑅)
5 rloc0g.3 . . . . 5 𝐿 = (𝑅 RLocal 𝑆)
6 rloc0g.4 . . . . 5 = (𝑅 ~RL 𝑆)
7 rloc0g.5 . . . . 5 (𝜑𝑅 ∈ CRing)
8 rloc0g.6 . . . . 5 (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
92, 3, 4, 5, 6, 7, 8rloccring 33256 . . . 4 (𝜑𝐿 ∈ CRing)
109crngringd 20263 . . 3 (𝜑𝐿 ∈ Ring)
11 eqid 2734 . . . . . . . . . 10 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1211, 2mgpbas 20157 . . . . . . . . 9 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
1312submss 18834 . . . . . . . 8 (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅))
148, 13syl 17 . . . . . . 7 (𝜑𝑆 ⊆ (Base‘𝑅))
15 rloc0g.2 . . . . . . . . . 10 1 = (1r𝑅)
1611, 15ringidval 20200 . . . . . . . . 9 1 = (0g‘(mulGrp‘𝑅))
1716subm0cl 18836 . . . . . . . 8 (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1𝑆)
188, 17syl 17 . . . . . . 7 (𝜑1𝑆)
1914, 18sseldd 3995 . . . . . 6 (𝜑1 ∈ (Base‘𝑅))
2019, 18opelxpd 5727 . . . . 5 (𝜑 → ⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆))
216ovexi 7464 . . . . . 6 ∈ V
2221ecelqsi 8811 . . . . 5 (⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆) → [⟨ 1 , 1 ⟩] ∈ (((Base‘𝑅) × 𝑆) / ))
2320, 22syl 17 . . . 4 (𝜑 → [⟨ 1 , 1 ⟩] ∈ (((Base‘𝑅) × 𝑆) / ))
24 rloc0g.1 . . . . 5 0 = (0g𝑅)
25 eqid 2734 . . . . 5 (-g𝑅) = (-g𝑅)
26 eqid 2734 . . . . 5 ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆)
272, 24, 3, 25, 26, 5, 6, 7, 14rlocbas 33253 . . . 4 (𝜑 → (((Base‘𝑅) × 𝑆) / ) = (Base‘𝐿))
2823, 27eleqtrd 2840 . . 3 (𝜑 → [⟨ 1 , 1 ⟩] ∈ (Base‘𝐿))
297ad4antr 732 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑅 ∈ CRing)
308ad4antr 732 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
3119ad4antr 732 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 1 ∈ (Base‘𝑅))
32 simpllr 776 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑎 ∈ (Base‘𝑅))
3330, 17syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 1𝑆)
34 simplr 769 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑏𝑆)
35 eqid 2734 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
362, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35rlocmulval 33255 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)[⟨𝑎, 𝑏⟩] ) = [⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩] )
3729crngringd 20263 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑅 ∈ Ring)
382, 3, 15, 37, 32ringlidmd 20285 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ( 1 (.r𝑅)𝑎) = 𝑎)
3930, 13syl 17 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑆 ⊆ (Base‘𝑅))
4039, 34sseldd 3995 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑏 ∈ (Base‘𝑅))
412, 3, 15, 37, 40ringlidmd 20285 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ( 1 (.r𝑅)𝑏) = 𝑏)
4238, 41opeq12d 4885 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4342eceq1d 8783 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → [⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩] = [⟨𝑎, 𝑏⟩] )
4436, 43eqtrd 2774 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)[⟨𝑎, 𝑏⟩] ) = [⟨𝑎, 𝑏⟩] )
45 simpr 484 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑥 = [⟨𝑎, 𝑏⟩] )
4645oveq2d 7446 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = ([⟨ 1 , 1 ⟩] (.r𝐿)[⟨𝑎, 𝑏⟩] ))
4744, 46, 453eqtr4d 2784 . . . . . 6 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥)
4827eqcomd 2740 . . . . . . . . 9 (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ))
4948eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐿) ↔ 𝑥 ∈ (((Base‘𝑅) × 𝑆) / )))
5049biimpa 476 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐿)) → 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ))
5150elrlocbasi 33252 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏𝑆 𝑥 = [⟨𝑎, 𝑏⟩] )
5247, 51r19.29vva 3213 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐿)) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥)
532, 3, 4, 5, 6, 29, 30, 32, 31, 34, 33, 35rlocmulval 33255 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨𝑎, 𝑏⟩] (.r𝐿)[⟨ 1 , 1 ⟩] ) = [⟨(𝑎(.r𝑅) 1 ), (𝑏(.r𝑅) 1 )⟩] )
542, 3, 15, 37, 32ringridmd 20286 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑎(.r𝑅) 1 ) = 𝑎)
552, 3, 15, 37, 40ringridmd 20286 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑏(.r𝑅) 1 ) = 𝑏)
5654, 55opeq12d 4885 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ⟨(𝑎(.r𝑅) 1 ), (𝑏(.r𝑅) 1 )⟩ = ⟨𝑎, 𝑏⟩)
5756eceq1d 8783 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → [⟨(𝑎(.r𝑅) 1 ), (𝑏(.r𝑅) 1 )⟩] = [⟨𝑎, 𝑏⟩] )
5853, 57eqtrd 2774 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨𝑎, 𝑏⟩] (.r𝐿)[⟨ 1 , 1 ⟩] ) = [⟨𝑎, 𝑏⟩] )
5945oveq1d 7445 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = ([⟨𝑎, 𝑏⟩] (.r𝐿)[⟨ 1 , 1 ⟩] ))
6058, 59, 453eqtr4d 2784 . . . . . 6 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥)
6160, 51r19.29vva 3213 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐿)) → (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥)
6252, 61jca 511 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐿)) → (([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥 ∧ (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥))
6362ralrimiva 3143 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥 ∧ (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥))
64 eqid 2734 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
65 eqid 2734 . . . . 5 (1r𝐿) = (1r𝐿)
6664, 35, 65isringid 20284 . . . 4 (𝐿 ∈ Ring → (([⟨ 1 , 1 ⟩] ∈ (Base‘𝐿) ∧ ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥 ∧ (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥)) ↔ (1r𝐿) = [⟨ 1 , 1 ⟩] ))
6766biimpa 476 . . 3 ((𝐿 ∈ Ring ∧ ([⟨ 1 , 1 ⟩] ∈ (Base‘𝐿) ∧ ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥 ∧ (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥))) → (1r𝐿) = [⟨ 1 , 1 ⟩] )
6810, 28, 63, 67syl12anc 837 . 2 (𝜑 → (1r𝐿) = [⟨ 1 , 1 ⟩] )
691, 68eqtr4id 2793 1 (𝜑𝐼 = (1r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wss 3962  cop 4636   × cxp 5686  cfv 6562  (class class class)co 7430  [cec 8741   / cqs 8742  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  0gc0g 17485  SubMndcsubmnd 18807  -gcsg 18965  mulGrpcmgp 20151  1rcur 20198  Ringcrg 20250  CRingccrg 20251   ~RL cerl 33239   RLocal crloc 33240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-ec 8745  df-qs 8749  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17487  df-imas 17554  df-qus 17555  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-erl 33241  df-rloc 33242
This theorem is referenced by:  rlocf1  33259  fracfld  33289  zringfrac  33561
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