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Theorem opprunitd 13421
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 2190 . . . . . 6 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜π‘…))
3 eqidd 2190 . . . . . 6 (πœ‘ β†’ (βˆ₯rβ€˜π‘…) = (βˆ₯rβ€˜π‘…))
4 opprunitd.2 . . . . . 6 (πœ‘ β†’ 𝑆 = (opprβ€˜π‘…))
5 eqidd 2190 . . . . . 6 (πœ‘ β†’ (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†))
6 opprunitd.r . . . . . . 7 (πœ‘ β†’ 𝑅 ∈ Ring)
7 ringsrg 13360 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
86, 7syl 14 . . . . . 6 (πœ‘ β†’ 𝑅 ∈ SRing)
91, 2, 3, 4, 5, 8isunitd 13417 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ∧ π‘₯(βˆ₯rβ€˜π‘†)(1rβ€˜π‘…))))
10 eqid 2189 . . . . . . . . . . . . . . 15 (opprβ€˜π‘…) = (opprβ€˜π‘…)
1110opprring 13390 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring β†’ (opprβ€˜π‘…) ∈ Ring)
126, 11syl 14 . . . . . . . . . . . . 13 (πœ‘ β†’ (opprβ€˜π‘…) ∈ Ring)
134, 12eqeltrd 2266 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆 ∈ Ring)
14 vex 2755 . . . . . . . . . . . . 13 𝑦 ∈ V
1514a1i 9 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑦 ∈ V)
16 vex 2755 . . . . . . . . . . . . 13 π‘₯ ∈ V
1716a1i 9 . . . . . . . . . . . 12 (πœ‘ β†’ π‘₯ ∈ V)
18 eqid 2189 . . . . . . . . . . . . 13 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
19 eqid 2189 . . . . . . . . . . . . 13 (.rβ€˜π‘†) = (.rβ€˜π‘†)
20 eqid 2189 . . . . . . . . . . . . 13 (opprβ€˜π‘†) = (opprβ€˜π‘†)
21 eqid 2189 . . . . . . . . . . . . 13 (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†))
2218, 19, 20, 21opprmulg 13382 . . . . . . . . . . . 12 ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ π‘₯ ∈ V) β†’ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (π‘₯(.rβ€˜π‘†)𝑦))
2313, 15, 17, 22syl3anc 1249 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (π‘₯(.rβ€˜π‘†)𝑦))
244fveq2d 5534 . . . . . . . . . . . 12 (πœ‘ β†’ (.rβ€˜π‘†) = (.rβ€˜(opprβ€˜π‘…)))
2524oveqd 5908 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯(.rβ€˜π‘†)𝑦) = (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦))
26 eqid 2189 . . . . . . . . . . . . 13 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
27 eqid 2189 . . . . . . . . . . . . 13 (.rβ€˜π‘…) = (.rβ€˜π‘…)
28 eqid 2189 . . . . . . . . . . . . 13 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
2926, 27, 10, 28opprmulg 13382 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
306, 17, 15, 29syl3anc 1249 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯(.rβ€˜(opprβ€˜π‘…))𝑦) = (𝑦(.rβ€˜π‘…)π‘₯))
3123, 25, 303eqtrrd 2227 . . . . . . . . . 10 (πœ‘ β†’ (𝑦(.rβ€˜π‘…)π‘₯) = (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯))
3231eqeq1d 2198 . . . . . . . . 9 (πœ‘ β†’ ((𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…) ↔ (𝑦(.rβ€˜(opprβ€˜π‘†))π‘₯) = (1rβ€˜π‘…)))
3332rexbidv 2491 . . . . . . . 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 2190 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘…))
36 eqidd 2190 . . . . . . . 8 (πœ‘ β†’ (.rβ€˜π‘…) = (.rβ€˜π‘…))
3735, 3, 8, 36dvdsrd 13405 . . . . . . 7 (πœ‘ β†’ (π‘₯(βˆ₯rβ€˜π‘…)(1rβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ βˆƒπ‘¦ ∈ (Baseβ€˜π‘…)(𝑦(.rβ€˜π‘…)π‘₯) = (1rβ€˜π‘…))))
3810, 26opprbasg 13386 . . . . . . . . . 10 (𝑅 ∈ SRing β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
398, 38syl 14 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘…)))
404fveq2d 5534 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘…)))
4120, 18opprbasg 13386 . . . . . . . . . 10 (𝑆 ∈ Ring β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
4213, 41syl 14 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
4339, 40, 423eqtr2d 2228 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜(opprβ€˜π‘†)))
44 eqidd 2190 . . . . . . . 8 (πœ‘ β†’ (βˆ₯rβ€˜(opprβ€˜π‘†)) = (βˆ₯rβ€˜(opprβ€˜π‘†)))
4520opprring 13390 . . . . . . . . . 10 (𝑆 ∈ Ring β†’ (opprβ€˜π‘†) ∈ Ring)
4613, 45syl 14 . . . . . . . . 9 (πœ‘ β†’ (opprβ€˜π‘†) ∈ Ring)
47 ringsrg 13360 . . . . . . . . 9 ((opprβ€˜π‘†) ∈ Ring β†’ (opprβ€˜π‘†) ∈ SRing)
4846, 47syl 14 . . . . . . . 8 (πœ‘ β†’ (opprβ€˜π‘†) ∈ SRing)
49 eqidd 2190 . . . . . . . 8 (πœ‘ β†’ (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†)))
5043, 44, 48, 49dvdsrd 13405 . . . . . . 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 2190 . . . 4 (πœ‘ β†’ (Unitβ€˜π‘†) = (Unitβ€˜π‘†))
56 eqid 2189 . . . . . . 7 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5710, 56oppr1g 13393 . . . . . 6 (𝑅 ∈ Ring β†’ (1rβ€˜π‘…) = (1rβ€˜(opprβ€˜π‘…)))
586, 57syl 14 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜(opprβ€˜π‘…)))
594fveq2d 5534 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜(opprβ€˜π‘…)))
6058, 59eqtr4d 2225 . . . 4 (πœ‘ β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
61 eqidd 2190 . . . 4 (πœ‘ β†’ (opprβ€˜π‘†) = (opprβ€˜π‘†))
62 ringsrg 13360 . . . . 5 (𝑆 ∈ Ring β†’ 𝑆 ∈ SRing)
6313, 62syl 14 . . . 4 (πœ‘ β†’ 𝑆 ∈ SRing)
6455, 60, 5, 61, 44, 63isunitd 13417 . . 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 2187 1 (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘†))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  βˆƒwrex 2469  Vcvv 2752   class class class wbr 4018  β€˜cfv 5231  (class class class)co 5891  Basecbs 12480  .rcmulr 12556  1rcur 13274  SRingcsrg 13278  Ringcrg 13311  opprcoppr 13378  βˆ₯rcdsr 13397  Unitcui 13398
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-pre-ltirr 7941  ax-pre-lttrn 7943  ax-pre-ltadd 7945
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-tpos 6264  df-pnf 8012  df-mnf 8013  df-ltxr 8015  df-inn 8938  df-2 8996  df-3 8997  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-plusg 12568  df-mulr 12569  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915  df-cmn 13186  df-abl 13187  df-mgp 13236  df-ur 13275  df-srg 13279  df-ring 13313  df-oppr 13379  df-dvdsr 13400  df-unit 13401
This theorem is referenced by: (None)
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