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Theorem rhmopp 30360
Description: A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
Assertion
Ref Expression
rhmopp (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))

Proof of Theorem rhmopp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . 2 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
2 eqid 2825 . 2 (1r‘(oppr𝑅)) = (1r‘(oppr𝑅))
3 eqid 2825 . 2 (1r‘(oppr𝑆)) = (1r‘(oppr𝑆))
4 eqid 2825 . 2 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
5 eqid 2825 . 2 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
6 rhmrcl1 19082 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7 eqid 2825 . . . 4 (oppr𝑅) = (oppr𝑅)
87opprringb 18993 . . 3 (𝑅 ∈ Ring ↔ (oppr𝑅) ∈ Ring)
96, 8sylib 210 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Ring)
10 rhmrcl2 19083 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11 eqid 2825 . . . 4 (oppr𝑆) = (oppr𝑆)
1211opprringb 18993 . . 3 (𝑆 ∈ Ring ↔ (oppr𝑆) ∈ Ring)
1310, 12sylib 210 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Ring)
14 eqid 2825 . . . . 5 (1r𝑅) = (1r𝑅)
157, 14oppr1 18995 . . . 4 (1r𝑅) = (1r‘(oppr𝑅))
1615eqcomi 2834 . . 3 (1r‘(oppr𝑅)) = (1r𝑅)
17 eqid 2825 . . . . 5 (1r𝑆) = (1r𝑆)
1811, 17oppr1 18995 . . . 4 (1r𝑆) = (1r‘(oppr𝑆))
1918eqcomi 2834 . . 3 (1r‘(oppr𝑆)) = (1r𝑆)
2016, 19rhm1 19093 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr𝑅))) = (1r‘(oppr𝑆)))
21 simpl 476 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22 simprr 789 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘(oppr𝑅)))
23 eqid 2825 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
247, 23opprbas 18990 . . . . 5 (Base‘𝑅) = (Base‘(oppr𝑅))
2522, 24syl6eleqr 2917 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26 simprl 787 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘(oppr𝑅)))
2726, 24syl6eleqr 2917 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28 eqid 2825 . . . . 5 (.r𝑅) = (.r𝑅)
29 eqid 2825 . . . . 5 (.r𝑆) = (.r𝑆)
3023, 28, 29rhmmul 19090 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3121, 25, 27, 30syl3anc 1494 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3223, 28, 7, 4opprmul 18987 . . . 4 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
3332fveq2i 6440 . . 3 (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = (𝐹‘(𝑦(.r𝑅)𝑥))
34 eqid 2825 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3534, 29, 11, 5opprmul 18987 . . 3 ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥))
3631, 33, 353eqtr4g 2886 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)))
37 ringgrp 18913 . . . . 5 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ Grp)
389, 37syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Grp)
39 ringgrp 18913 . . . . 5 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ Grp)
4013, 39syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Grp)
4123, 34rhmf 19089 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42 rhmghm 19088 . . . . . . . . 9 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4342ad2antrr 717 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44 simplr 785 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
45 simpr 479 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
46 eqid 2825 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
47 eqid 2825 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
4823, 46, 47ghmlin 18023 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
4943, 44, 45, 48syl3anc 1494 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5049ralrimiva 3175 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5150ralrimiva 3175 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5241, 51jca 507 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
5338, 40, 52jca31 510 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → (((oppr𝑅) ∈ Grp ∧ (oppr𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
5411, 34opprbas 18990 . . . 4 (Base‘𝑆) = (Base‘(oppr𝑆))
557, 46oppradd 18991 . . . 4 (+g𝑅) = (+g‘(oppr𝑅))
5611, 47oppradd 18991 . . . 4 (+g𝑆) = (+g‘(oppr𝑆))
5724, 54, 55, 56isghm 18018 . . 3 (𝐹 ∈ ((oppr𝑅) GrpHom (oppr𝑆)) ↔ (((oppr𝑅) ∈ Grp ∧ (oppr𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
5853, 57sylibr 226 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) GrpHom (oppr𝑆)))
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 19091 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  wf 6123  cfv 6127  (class class class)co 6910  Basecbs 16229  +gcplusg 16312  .rcmulr 16313  Grpcgrp 17783   GrpHom cghm 18015  1rcur 18862  Ringcrg 18908  opprcoppr 18983   RingHom crh 19075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-tpos 7622  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-plusg 16325  df-mulr 16326  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-mhm 17695  df-grp 17786  df-ghm 18016  df-mgp 18851  df-ur 18863  df-ring 18910  df-oppr 18984  df-rnghom 19078
This theorem is referenced by:  elrhmunit  30361
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