ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  crngunit GIF version

Theorem crngunit 13280
Description: Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
crngunit.1 π‘ˆ = (Unitβ€˜π‘…)
crngunit.2 1 = (1rβ€˜π‘…)
crngunit.3 βˆ₯ = (βˆ₯rβ€˜π‘…)
Assertion
Ref Expression
crngunit (𝑅 ∈ CRing β†’ (𝑋 ∈ π‘ˆ ↔ 𝑋 βˆ₯ 1 ))

Proof of Theorem crngunit
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 crngunit.1 . . . . 5 π‘ˆ = (Unitβ€˜π‘…)
21a1i 9 . . . 4 (𝑅 ∈ CRing β†’ π‘ˆ = (Unitβ€˜π‘…))
3 crngunit.2 . . . . 5 1 = (1rβ€˜π‘…)
43a1i 9 . . . 4 (𝑅 ∈ CRing β†’ 1 = (1rβ€˜π‘…))
5 crngunit.3 . . . . 5 βˆ₯ = (βˆ₯rβ€˜π‘…)
65a1i 9 . . . 4 (𝑅 ∈ CRing β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))
7 eqidd 2178 . . . 4 (𝑅 ∈ CRing β†’ (opprβ€˜π‘…) = (opprβ€˜π‘…))
8 eqidd 2178 . . . 4 (𝑅 ∈ CRing β†’ (βˆ₯rβ€˜(opprβ€˜π‘…)) = (βˆ₯rβ€˜(opprβ€˜π‘…)))
9 crngring 13191 . . . . 5 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
10 ringsrg 13224 . . . . 5 (𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
119, 10syl 14 . . . 4 (𝑅 ∈ CRing β†’ 𝑅 ∈ SRing)
122, 4, 6, 7, 8, 11isunitd 13275 . . 3 (𝑅 ∈ CRing β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 )))
13 eqid 2177 . . . . . . . . . . . 12 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
14 eqid 2177 . . . . . . . . . . . 12 (.rβ€˜π‘…) = (.rβ€˜π‘…)
15 eqid 2177 . . . . . . . . . . . 12 (opprβ€˜π‘…) = (opprβ€˜π‘…)
16 eqid 2177 . . . . . . . . . . . 12 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
1713, 14, 15, 16crngoppr 13244 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Baseβ€˜π‘…) ∧ 𝑋 ∈ (Baseβ€˜π‘…)) β†’ (𝑦(.rβ€˜π‘…)𝑋) = (𝑦(.rβ€˜(opprβ€˜π‘…))𝑋))
18173expa 1203 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Baseβ€˜π‘…)) ∧ 𝑋 ∈ (Baseβ€˜π‘…)) β†’ (𝑦(.rβ€˜π‘…)𝑋) = (𝑦(.rβ€˜(opprβ€˜π‘…))𝑋))
1918eqcomd 2183 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Baseβ€˜π‘…)) ∧ 𝑋 ∈ (Baseβ€˜π‘…)) β†’ (𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = (𝑦(.rβ€˜π‘…)𝑋))
2019an32s 568 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Baseβ€˜π‘…)) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ (𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = (𝑦(.rβ€˜π‘…)𝑋))
2120eqeq1d 2186 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Baseβ€˜π‘…)) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ ((𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = 1 ↔ (𝑦(.rβ€˜π‘…)𝑋) = 1 ))
2221rexbidva 2474 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Baseβ€˜π‘…)) β†’ (βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = 1 ↔ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)𝑋) = 1 ))
2322pm5.32da 452 . . . . 5 (𝑅 ∈ CRing β†’ ((𝑋 ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = 1 ) ↔ (𝑋 ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)𝑋) = 1 )))
2415, 13opprbasg 13247 . . . . . 6 (𝑅 ∈ CRing β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
2515opprring 13249 . . . . . . 7 (𝑅 ∈ Ring β†’ (opprβ€˜π‘…) ∈ Ring)
26 ringsrg 13224 . . . . . . 7 ((opprβ€˜π‘…) ∈ Ring β†’ (opprβ€˜π‘…) ∈ SRing)
279, 25, 263syl 17 . . . . . 6 (𝑅 ∈ CRing β†’ (opprβ€˜π‘…) ∈ SRing)
28 eqidd 2178 . . . . . 6 (𝑅 ∈ CRing β†’ (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…)))
2924, 8, 27, 28dvdsrd 13263 . . . . 5 (𝑅 ∈ CRing β†’ (𝑋(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ↔ (𝑋 ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘…))𝑋) = 1 )))
30 eqidd 2178 . . . . . 6 (𝑅 ∈ CRing β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘…))
31 eqidd 2178 . . . . . 6 (𝑅 ∈ CRing β†’ (.rβ€˜π‘…) = (.rβ€˜π‘…))
3230, 6, 11, 31dvdsrd 13263 . . . . 5 (𝑅 ∈ CRing β†’ (𝑋 βˆ₯ 1 ↔ (𝑋 ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)𝑋) = 1 )))
3323, 29, 323bitr4d 220 . . . 4 (𝑅 ∈ CRing β†’ (𝑋(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ↔ 𝑋 βˆ₯ 1 ))
3433anbi2d 464 . . 3 (𝑅 ∈ CRing β†’ ((𝑋 βˆ₯ 1 ∧ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ) ↔ (𝑋 βˆ₯ 1 ∧ 𝑋 βˆ₯ 1 )))
3512, 34bitrd 188 . 2 (𝑅 ∈ CRing β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋 βˆ₯ 1 )))
36 pm4.24 395 . 2 (𝑋 βˆ₯ 1 ↔ (𝑋 βˆ₯ 1 ∧ 𝑋 βˆ₯ 1 ))
3735, 36bitr4di 198 1 (𝑅 ∈ CRing β†’ (𝑋 ∈ π‘ˆ ↔ 𝑋 βˆ₯ 1 ))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   class class class wbr 4004  β€˜cfv 5217  (class class class)co 5875  Basecbs 12462  .rcmulr 12537  1rcur 13142  SRingcsrg 13146  Ringcrg 13179  CRingccrg 13180  opprcoppr 13239  βˆ₯rcdsr 13255  Unitcui 13256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-tpos 6246  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-cmn 13090  df-abl 13091  df-mgp 13131  df-ur 13143  df-srg 13147  df-ring 13181  df-cring 13182  df-oppr 13240  df-dvdsr 13258  df-unit 13259
This theorem is referenced by:  dvdsunit  13281
  Copyright terms: Public domain W3C validator