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Theorem opprunitd 13277
Description: Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
opprunitd.1 (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))
opprunitd.2 (πœ‘ β†’ 𝑆 = (opprβ€˜π‘…))
opprunitd.r (πœ‘ β†’ 𝑅 ∈ Ring)
Assertion
Ref Expression
opprunitd (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘†))

Proof of Theorem opprunitd
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprunitd.1 . . . . . 6 (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))
2 eqidd 2178 . . . . . 6 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜π‘…))
3 eqidd 2178 . . . . . 6 (πœ‘ β†’ (βˆ₯rβ€˜π‘…) = (βˆ₯rβ€˜π‘…))
4 opprunitd.2 . . . . . 6 (πœ‘ β†’ 𝑆 = (opprβ€˜π‘…))
5 eqidd 2178 . . . . . 6 (πœ‘ β†’ (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†))
6 opprunitd.r . . . . . . 7 (πœ‘ β†’ 𝑅 ∈ Ring)
7 ringsrg 13222 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
86, 7syl 14 . . . . . 6 (πœ‘ β†’ 𝑅 ∈ SRing)
91, 2, 3, 4, 5, 8isunitd 13273 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…))))
10 eqid 2177 . . . . . . . . . . . . . . 15 (opprβ€˜π‘…) = (opprβ€˜π‘…)
1110opprring 13247 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring β†’ (opprβ€˜π‘…) ∈ Ring)
126, 11syl 14 . . . . . . . . . . . . 13 (πœ‘ β†’ (opprβ€˜π‘…) ∈ Ring)
134, 12eqeltrd 2254 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆 ∈ Ring)
14 vex 2740 . . . . . . . . . . . . 13 𝑦 ∈ V
1514a1i 9 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑦 ∈ V)
16 vex 2740 . . . . . . . . . . . . 13 π‘₯ ∈ V
1716a1i 9 . . . . . . . . . . . 12 (πœ‘ β†’ π‘₯ ∈ V)
18 eqid 2177 . . . . . . . . . . . . 13 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
19 eqid 2177 . . . . . . . . . . . . 13 (.rβ€˜π‘†) = (.rβ€˜π‘†)
20 eqid 2177 . . . . . . . . . . . . 13 (opprβ€˜π‘†) = (opprβ€˜π‘†)
21 eqid 2177 . . . . . . . . . . . . 13 (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†))
2218, 19, 20, 21opprmulg 13241 . . . . . . . . . . . 12 ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ π‘₯ ∈ V) β†’ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (π‘₯(.rβ€˜π‘†)𝑦))
2313, 15, 17, 22syl3anc 1238 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (π‘₯(.rβ€˜π‘†)𝑦))
244fveq2d 5519 . . . . . . . . . . . 12 (πœ‘ β†’ (.rβ€˜π‘†) = (.rβ€˜(opprβ€˜π‘…)))
2524oveqd 5891 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯(.rβ€˜π‘†)𝑦) = (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦))
26 eqid 2177 . . . . . . . . . . . . 13 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
27 eqid 2177 . . . . . . . . . . . . 13 (.rβ€˜π‘…) = (.rβ€˜π‘…)
28 eqid 2177 . . . . . . . . . . . . 13 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
2926, 27, 10, 28opprmulg 13241 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
306, 17, 15, 29syl3anc 1238 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
3123, 25, 303eqtrrd 2215 . . . . . . . . . 10 (πœ‘ β†’ (𝑦(.rβ€˜π‘…)π‘₯) = (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯))
3231eqeq1d 2186 . . . . . . . . 9 (πœ‘ β†’ ((𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…) ↔ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (1rβ€˜π‘…)))
3332rexbidv 2478 . . . . . . . 8 (πœ‘ β†’ (βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…) ↔ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (1rβ€˜π‘…)))
3433anbi2d 464 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…)) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (1rβ€˜π‘…))))
35 eqidd 2178 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘…))
36 eqidd 2178 . . . . . . . 8 (πœ‘ β†’ (.rβ€˜π‘…) = (.rβ€˜π‘…))
3735, 3, 8, 36dvdsrd 13261 . . . . . . 7 (πœ‘ β†’ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…))))
3810, 26opprbasg 13245 . . . . . . . . . 10 (𝑅 ∈ SRing β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
398, 38syl 14 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
404fveq2d 5519 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘…)))
4120, 18opprbasg 13245 . . . . . . . . . 10 (𝑆 ∈ Ring β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
4213, 41syl 14 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
4339, 40, 423eqtr2d 2216 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘†)))
44 eqidd 2178 . . . . . . . 8 (πœ‘ β†’ (βˆ₯rβ€˜(opprβ€˜π‘†)) = (βˆ₯rβ€˜(opprβ€˜π‘†)))
4520opprring 13247 . . . . . . . . . 10 (𝑆 ∈ Ring β†’ (opprβ€˜π‘†) ∈ Ring)
4613, 45syl 14 . . . . . . . . 9 (πœ‘ β†’ (opprβ€˜π‘†) ∈ Ring)
47 ringsrg 13222 . . . . . . . . 9 ((opprβ€˜π‘†) ∈ Ring β†’ (opprβ€˜π‘†) ∈ SRing)
4846, 47syl 14 . . . . . . . 8 (πœ‘ β†’ (opprβ€˜π‘†) ∈ SRing)
49 eqidd 2178 . . . . . . . 8 (πœ‘ β†’ (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†)))
5043, 44, 48, 49dvdsrd 13261 . . . . . . 7 (πœ‘ β†’ (π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (1rβ€˜π‘…))))
5134, 37, 503bitr4d 220 . . . . . 6 (πœ‘ β†’ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ↔ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…)))
5251anbi1d 465 . . . . 5 (πœ‘ β†’ ((π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…)) ↔ (π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…))))
539, 52bitrd 188 . . . 4 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…))))
5453biancomd 271 . . 3 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…))))
55 eqidd 2178 . . . 4 (πœ‘ β†’ (Unitβ€˜π‘†) = (Unitβ€˜π‘†))
56 eqid 2177 . . . . . . 7 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5710, 56oppr1g 13250 . . . . . 6 (𝑅 ∈ Ring β†’ (1rβ€˜π‘…) = (1rβ€˜(opprβ€˜π‘…)))
586, 57syl 14 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜(opprβ€˜π‘…)))
594fveq2d 5519 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜(opprβ€˜π‘…)))
6058, 59eqtr4d 2213 . . . 4 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
61 eqidd 2178 . . . 4 (πœ‘ β†’ (opprβ€˜π‘†) = (opprβ€˜π‘†))
62 ringsrg 13222 . . . . 5 (𝑆 ∈ Ring β†’ 𝑆 ∈ SRing)
6313, 62syl 14 . . . 4 (πœ‘ β†’ 𝑆 ∈ SRing)
6455, 60, 5, 61, 44, 63isunitd 13273 . . 3 (πœ‘ β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘…))))
6554, 64bitr4d 191 . 2 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↔ π‘₯ ∈ (Unitβ€˜π‘†)))
6665eqrdv 2175 1 (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘†))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456  Vcvv 2737   class class class wbr 4003  β€˜cfv 5216  (class class class)co 5874  Basecbs 12461  .rcmulr 12536  1rcur 13140  SRingcsrg 13144  Ringcrg 13177  opprcoppr 13237  βˆ₯rcdsr 13253  Unitcui 13254
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7993  df-mnf 7994  df-ltxr 7996  df-inn 8919  df-2 8977  df-3 8978  df-ndx 12464  df-slot 12465  df-base 12467  df-sets 12468  df-plusg 12548  df-mulr 12549  df-0g 12706  df-mgm 12774  df-sgrp 12807  df-mnd 12817  df-grp 12879  df-minusg 12880  df-cmn 13088  df-abl 13089  df-mgp 13129  df-ur 13141  df-srg 13145  df-ring 13179  df-oppr 13238  df-dvdsr 13256  df-unit 13257
This theorem is referenced by: (None)
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