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Theorem rhmopp 30159
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 2771 . 2 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
2 eqid 2771 . 2 (1r‘(oppr𝑅)) = (1r‘(oppr𝑅))
3 eqid 2771 . 2 (1r‘(oppr𝑆)) = (1r‘(oppr𝑆))
4 eqid 2771 . 2 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
5 eqid 2771 . 2 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
6 rhmrcl1 18929 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7 eqid 2771 . . . 4 (oppr𝑅) = (oppr𝑅)
87opprringb 18840 . . 3 (𝑅 ∈ Ring ↔ (oppr𝑅) ∈ Ring)
96, 8sylib 208 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Ring)
10 rhmrcl2 18930 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11 eqid 2771 . . . 4 (oppr𝑆) = (oppr𝑆)
1211opprringb 18840 . . 3 (𝑆 ∈ Ring ↔ (oppr𝑆) ∈ Ring)
1310, 12sylib 208 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Ring)
14 eqid 2771 . . . . 5 (1r𝑅) = (1r𝑅)
157, 14oppr1 18842 . . . 4 (1r𝑅) = (1r‘(oppr𝑅))
1615eqcomi 2780 . . 3 (1r‘(oppr𝑅)) = (1r𝑅)
17 eqid 2771 . . . . 5 (1r𝑆) = (1r𝑆)
1811, 17oppr1 18842 . . . 4 (1r𝑆) = (1r‘(oppr𝑆))
1918eqcomi 2780 . . 3 (1r‘(oppr𝑆)) = (1r𝑆)
2016, 19rhm1 18940 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr𝑅))) = (1r‘(oppr𝑆)))
21 simpl 468 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22 simprr 756 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘(oppr𝑅)))
23 eqid 2771 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
247, 23opprbas 18837 . . . . 5 (Base‘𝑅) = (Base‘(oppr𝑅))
2522, 24syl6eleqr 2861 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26 simprl 754 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘(oppr𝑅)))
2726, 24syl6eleqr 2861 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28 eqid 2771 . . . . 5 (.r𝑅) = (.r𝑅)
29 eqid 2771 . . . . 5 (.r𝑆) = (.r𝑆)
3023, 28, 29rhmmul 18937 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3121, 25, 27, 30syl3anc 1476 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3223, 28, 7, 4opprmul 18834 . . . 4 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
3332fveq2i 6336 . . 3 (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = (𝐹‘(𝑦(.r𝑅)𝑥))
34 eqid 2771 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3534, 29, 11, 5opprmul 18834 . . 3 ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥))
3631, 33, 353eqtr4g 2830 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)))
37 ringgrp 18760 . . . . 5 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ Grp)
389, 37syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Grp)
39 ringgrp 18760 . . . . 5 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ Grp)
4013, 39syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Grp)
4123, 34rhmf 18936 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42 rhmghm 18935 . . . . . . . . 9 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4342ad2antrr 705 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44 simplr 752 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
45 simpr 471 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
46 eqid 2771 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
47 eqid 2771 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
4823, 46, 47ghmlin 17873 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
4943, 44, 45, 48syl3anc 1476 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5049ralrimiva 3115 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5150ralrimiva 3115 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5241, 51jca 501 . . . 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 504 . . 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 18837 . . . 4 (Base‘𝑆) = (Base‘(oppr𝑆))
557, 46oppradd 18838 . . . 4 (+g𝑅) = (+g‘(oppr𝑅))
5611, 47oppradd 18838 . . . 4 (+g𝑆) = (+g‘(oppr𝑆))
5724, 54, 55, 56isghm 17868 . . 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 224 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) GrpHom (oppr𝑆)))
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 18938 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wf 6026  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  .rcmulr 16150  Grpcgrp 17630   GrpHom cghm 17865  1rcur 18709  Ringcrg 18755  opprcoppr 18830   RingHom crh 18922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-tpos 7508  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-plusg 16162  df-mulr 16163  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-grp 17633  df-ghm 17866  df-mgp 18698  df-ur 18710  df-ring 18757  df-oppr 18831  df-rnghom 18925
This theorem is referenced by:  elrhmunit  30160
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