Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rloc1r Structured version   Visualization version   GIF version

Theorem rloc1r 33276
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 2737 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 eqid 2737 . . . . 5 (.r𝑅) = (.r𝑅)
4 eqid 2737 . . . . 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 33274 . . . 4 (𝜑𝐿 ∈ CRing)
109crngringd 20243 . . 3 (𝜑𝐿 ∈ Ring)
11 eqid 2737 . . . . . . . . . 10 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1211, 2mgpbas 20142 . . . . . . . . 9 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
1312submss 18822 . . . . . . . 8 (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅))
148, 13syl 17 . . . . . . 7 (𝜑𝑆 ⊆ (Base‘𝑅))
15 rloc0g.2 . . . . . . . . . 10 1 = (1r𝑅)
1611, 15ringidval 20180 . . . . . . . . 9 1 = (0g‘(mulGrp‘𝑅))
1716subm0cl 18824 . . . . . . . 8 (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1𝑆)
188, 17syl 17 . . . . . . 7 (𝜑1𝑆)
1914, 18sseldd 3984 . . . . . 6 (𝜑1 ∈ (Base‘𝑅))
2019, 18opelxpd 5724 . . . . 5 (𝜑 → ⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆))
216ovexi 7465 . . . . . 6 ∈ V
2221ecelqsi 8813 . . . . 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 2737 . . . . 5 (-g𝑅) = (-g𝑅)
26 eqid 2737 . . . . 5 ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆)
272, 24, 3, 25, 26, 5, 6, 7, 14rlocbas 33271 . . . 4 (𝜑 → (((Base‘𝑅) × 𝑆) / ) = (Base‘𝐿))
2823, 27eleqtrd 2843 . . 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 2737 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
362, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35rlocmulval 33273 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)[⟨𝑎, 𝑏⟩] ) = [⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩] )
3729crngringd 20243 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑅 ∈ Ring)
382, 3, 15, 37, 32ringlidmd 20269 . . . . . . . . . 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 3984 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → 𝑏 ∈ (Base‘𝑅))
412, 3, 15, 37, 40ringlidmd 20269 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ( 1 (.r𝑅)𝑏) = 𝑏)
4238, 41opeq12d 4881 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4342eceq1d 8785 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → [⟨( 1 (.r𝑅)𝑎), ( 1 (.r𝑅)𝑏)⟩] = [⟨𝑎, 𝑏⟩] )
4436, 43eqtrd 2777 . . . . . . 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 7447 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = ([⟨ 1 , 1 ⟩] (.r𝐿)[⟨𝑎, 𝑏⟩] ))
4744, 46, 453eqtr4d 2787 . . . . . 6 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥)
4827eqcomd 2743 . . . . . . . . 9 (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ))
4948eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐿) ↔ 𝑥 ∈ (((Base‘𝑅) × 𝑆) / )))
5049biimpa 476 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐿)) → 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ))
5150elrlocbasi 33270 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏𝑆 𝑥 = [⟨𝑎, 𝑏⟩] )
5247, 51r19.29vva 3216 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐿)) → ([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥)
532, 3, 4, 5, 6, 29, 30, 32, 31, 34, 33, 35rlocmulval 33273 . . . . . . . 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 20270 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑎(.r𝑅) 1 ) = 𝑎)
552, 3, 15, 37, 40ringridmd 20270 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑏(.r𝑅) 1 ) = 𝑏)
5654, 55opeq12d 4881 . . . . . . . . 9 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ⟨(𝑎(.r𝑅) 1 ), (𝑏(.r𝑅) 1 )⟩ = ⟨𝑎, 𝑏⟩)
5756eceq1d 8785 . . . . . . . 8 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → [⟨(𝑎(.r𝑅) 1 ), (𝑏(.r𝑅) 1 )⟩] = [⟨𝑎, 𝑏⟩] )
5853, 57eqtrd 2777 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → ([⟨𝑎, 𝑏⟩] (.r𝐿)[⟨ 1 , 1 ⟩] ) = [⟨𝑎, 𝑏⟩] )
5945oveq1d 7446 . . . . . . 7 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = ([⟨𝑎, 𝑏⟩] (.r𝐿)[⟨ 1 , 1 ⟩] ))
6058, 59, 453eqtr4d 2787 . . . . . 6 (((((𝜑𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ) → (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥)
6160, 51r19.29vva 3216 . . . . 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 3146 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] (.r𝐿)𝑥) = 𝑥 ∧ (𝑥(.r𝐿)[⟨ 1 , 1 ⟩] ) = 𝑥))
64 eqid 2737 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
65 eqid 2737 . . . . 5 (1r𝐿) = (1r𝐿)
6664, 35, 65isringid 20268 . . . 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 2796 1 (𝜑𝐼 = (1r𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484  SubMndcsubmnd 18795  -gcsg 18953  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  CRingccrg 20231   ~RL cerl 33257   RLocal crloc 33258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-erl 33259  df-rloc 33260
This theorem is referenced by:  rlocf1  33277  fracfld  33310  zringfrac  33582
  Copyright terms: Public domain W3C validator