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Theorem opprunitd 13606
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 (𝜑𝑈 = (Unit‘𝑅))
opprunitd.2 (𝜑𝑆 = (oppr𝑅))
opprunitd.r (𝜑𝑅 ∈ Ring)
Assertion
Ref Expression
opprunitd (𝜑𝑈 = (Unit‘𝑆))

Proof of Theorem opprunitd
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprunitd.1 . . . . . 6 (𝜑𝑈 = (Unit‘𝑅))
2 eqidd 2194 . . . . . 6 (𝜑 → (1r𝑅) = (1r𝑅))
3 eqidd 2194 . . . . . 6 (𝜑 → (∥r𝑅) = (∥r𝑅))
4 opprunitd.2 . . . . . 6 (𝜑𝑆 = (oppr𝑅))
5 eqidd 2194 . . . . . 6 (𝜑 → (∥r𝑆) = (∥r𝑆))
6 opprunitd.r . . . . . . 7 (𝜑𝑅 ∈ Ring)
7 ringsrg 13543 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ SRing)
86, 7syl 14 . . . . . 6 (𝜑𝑅 ∈ SRing)
91, 2, 3, 4, 5, 8isunitd 13602 . . . . 5 (𝜑 → (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r𝑆)(1r𝑅))))
10 eqid 2193 . . . . . . . . . . . . . . 15 (oppr𝑅) = (oppr𝑅)
1110opprring 13575 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
126, 11syl 14 . . . . . . . . . . . . 13 (𝜑 → (oppr𝑅) ∈ Ring)
134, 12eqeltrd 2270 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Ring)
14 vex 2763 . . . . . . . . . . . . 13 𝑦 ∈ V
1514a1i 9 . . . . . . . . . . . 12 (𝜑𝑦 ∈ V)
16 vex 2763 . . . . . . . . . . . . 13 𝑥 ∈ V
1716a1i 9 . . . . . . . . . . . 12 (𝜑𝑥 ∈ V)
18 eqid 2193 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
19 eqid 2193 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
20 eqid 2193 . . . . . . . . . . . . 13 (oppr𝑆) = (oppr𝑆)
21 eqid 2193 . . . . . . . . . . . . 13 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
2218, 19, 20, 21opprmulg 13567 . . . . . . . . . . . 12 ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘(oppr𝑆))𝑥) = (𝑥(.r𝑆)𝑦))
2313, 15, 17, 22syl3anc 1249 . . . . . . . . . . 11 (𝜑 → (𝑦(.r‘(oppr𝑆))𝑥) = (𝑥(.r𝑆)𝑦))
244fveq2d 5558 . . . . . . . . . . . 12 (𝜑 → (.r𝑆) = (.r‘(oppr𝑅)))
2524oveqd 5935 . . . . . . . . . . 11 (𝜑 → (𝑥(.r𝑆)𝑦) = (𝑥(.r‘(oppr𝑅))𝑦))
26 eqid 2193 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
27 eqid 2193 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
28 eqid 2193 . . . . . . . . . . . . 13 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
2926, 27, 10, 28opprmulg 13567 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥))
306, 17, 15, 29syl3anc 1249 . . . . . . . . . . 11 (𝜑 → (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥))
3123, 25, 303eqtrrd 2231 . . . . . . . . . 10 (𝜑 → (𝑦(.r𝑅)𝑥) = (𝑦(.r‘(oppr𝑆))𝑥))
3231eqeq1d 2202 . . . . . . . . 9 (𝜑 → ((𝑦(.r𝑅)𝑥) = (1r𝑅) ↔ (𝑦(.r‘(oppr𝑆))𝑥) = (1r𝑅)))
3332rexbidv 2495 . . . . . . . 8 (𝜑 → (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr𝑆))𝑥) = (1r𝑅)))
3433anbi2d 464 . . . . . . 7 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr𝑆))𝑥) = (1r𝑅))))
35 eqidd 2194 . . . . . . . 8 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
36 eqidd 2194 . . . . . . . 8 (𝜑 → (.r𝑅) = (.r𝑅))
3735, 3, 8, 36dvdsrd 13590 . . . . . . 7 (𝜑 → (𝑥(∥r𝑅)(1r𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅))))
3810, 26opprbasg 13571 . . . . . . . . . 10 (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘(oppr𝑅)))
398, 38syl 14 . . . . . . . . 9 (𝜑 → (Base‘𝑅) = (Base‘(oppr𝑅)))
404fveq2d 5558 . . . . . . . . 9 (𝜑 → (Base‘𝑆) = (Base‘(oppr𝑅)))
4120, 18opprbasg 13571 . . . . . . . . . 10 (𝑆 ∈ Ring → (Base‘𝑆) = (Base‘(oppr𝑆)))
4213, 41syl 14 . . . . . . . . 9 (𝜑 → (Base‘𝑆) = (Base‘(oppr𝑆)))
4339, 40, 423eqtr2d 2232 . . . . . . . 8 (𝜑 → (Base‘𝑅) = (Base‘(oppr𝑆)))
44 eqidd 2194 . . . . . . . 8 (𝜑 → (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆)))
4520opprring 13575 . . . . . . . . . 10 (𝑆 ∈ Ring → (oppr𝑆) ∈ Ring)
4613, 45syl 14 . . . . . . . . 9 (𝜑 → (oppr𝑆) ∈ Ring)
47 ringsrg 13543 . . . . . . . . 9 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ SRing)
4846, 47syl 14 . . . . . . . 8 (𝜑 → (oppr𝑆) ∈ SRing)
49 eqidd 2194 . . . . . . . 8 (𝜑 → (.r‘(oppr𝑆)) = (.r‘(oppr𝑆)))
5043, 44, 48, 49dvdsrd 13590 . . . . . . 7 (𝜑 → (𝑥(∥r‘(oppr𝑆))(1r𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr𝑆))𝑥) = (1r𝑅))))
5134, 37, 503bitr4d 220 . . . . . 6 (𝜑 → (𝑥(∥r𝑅)(1r𝑅) ↔ 𝑥(∥r‘(oppr𝑆))(1r𝑅)))
5251anbi1d 465 . . . . 5 (𝜑 → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r𝑆)(1r𝑅)) ↔ (𝑥(∥r‘(oppr𝑆))(1r𝑅) ∧ 𝑥(∥r𝑆)(1r𝑅))))
539, 52bitrd 188 . . . 4 (𝜑 → (𝑥𝑈 ↔ (𝑥(∥r‘(oppr𝑆))(1r𝑅) ∧ 𝑥(∥r𝑆)(1r𝑅))))
5453biancomd 271 . . 3 (𝜑 → (𝑥𝑈 ↔ (𝑥(∥r𝑆)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑅))))
55 eqidd 2194 . . . 4 (𝜑 → (Unit‘𝑆) = (Unit‘𝑆))
56 eqid 2193 . . . . . . 7 (1r𝑅) = (1r𝑅)
5710, 56oppr1g 13578 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) = (1r‘(oppr𝑅)))
586, 57syl 14 . . . . 5 (𝜑 → (1r𝑅) = (1r‘(oppr𝑅)))
594fveq2d 5558 . . . . 5 (𝜑 → (1r𝑆) = (1r‘(oppr𝑅)))
6058, 59eqtr4d 2229 . . . 4 (𝜑 → (1r𝑅) = (1r𝑆))
61 eqidd 2194 . . . 4 (𝜑 → (oppr𝑆) = (oppr𝑆))
62 ringsrg 13543 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ SRing)
6313, 62syl 14 . . . 4 (𝜑𝑆 ∈ SRing)
6455, 60, 5, 61, 44, 63isunitd 13602 . . 3 (𝜑 → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥(∥r𝑆)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑆))(1r𝑅))))
6554, 64bitr4d 191 . 2 (𝜑 → (𝑥𝑈𝑥 ∈ (Unit‘𝑆)))
6665eqrdv 2191 1 (𝜑𝑈 = (Unit‘𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  wrex 2473  Vcvv 2760   class class class wbr 4029  cfv 5254  (class class class)co 5918  Basecbs 12618  .rcmulr 12696  1rcur 13455  SRingcsrg 13459  Ringcrg 13492  opprcoppr 13563  rcdsr 13582  Unitcui 13583
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586
This theorem is referenced by: (None)
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