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Theorem rhmopp 30808
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 19393 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7 eqid 2825 . . . 4 (oppr𝑅) = (oppr𝑅)
87opprringb 19304 . . 3 (𝑅 ∈ Ring ↔ (oppr𝑅) ∈ Ring)
96, 8sylib 219 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Ring)
10 rhmrcl2 19394 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11 eqid 2825 . . . 4 (oppr𝑆) = (oppr𝑆)
1211opprringb 19304 . . 3 (𝑆 ∈ Ring ↔ (oppr𝑆) ∈ Ring)
1310, 12sylib 219 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Ring)
14 eqid 2825 . . . . 5 (1r𝑅) = (1r𝑅)
157, 14oppr1 19306 . . . 4 (1r𝑅) = (1r‘(oppr𝑅))
1615eqcomi 2834 . . 3 (1r‘(oppr𝑅)) = (1r𝑅)
17 eqid 2825 . . . . 5 (1r𝑆) = (1r𝑆)
1811, 17oppr1 19306 . . . 4 (1r𝑆) = (1r‘(oppr𝑆))
1918eqcomi 2834 . . 3 (1r‘(oppr𝑆)) = (1r𝑆)
2016, 19rhm1 19404 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr𝑅))) = (1r‘(oppr𝑆)))
21 simpl 483 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22 simprr 769 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘(oppr𝑅)))
23 eqid 2825 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
247, 23opprbas 19301 . . . . 5 (Base‘𝑅) = (Base‘(oppr𝑅))
2522, 24syl6eleqr 2928 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26 simprl 767 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘(oppr𝑅)))
2726, 24syl6eleqr 2928 . . . 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 19401 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3121, 25, 27, 30syl3anc 1365 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3223, 28, 7, 4opprmul 19298 . . . 4 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
3332fveq2i 6669 . . 3 (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = (𝐹‘(𝑦(.r𝑅)𝑥))
34 eqid 2825 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3534, 29, 11, 5opprmul 19298 . . 3 ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥))
3631, 33, 353eqtr4g 2885 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)))
37 ringgrp 19224 . . . . 5 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ Grp)
389, 37syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Grp)
39 ringgrp 19224 . . . . 5 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ Grp)
4013, 39syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Grp)
4123, 34rhmf 19400 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42 rhmghm 19399 . . . . . . . . 9 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4342ad2antrr 722 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44 simplr 765 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
45 simpr 485 . . . . . . . 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 18295 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
4943, 44, 45, 48syl3anc 1365 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5049ralrimiva 3186 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5150ralrimiva 3186 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5241, 51jca 512 . . . 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 515 . . 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 19301 . . . 4 (Base‘𝑆) = (Base‘(oppr𝑆))
557, 46oppradd 19302 . . . 4 (+g𝑅) = (+g‘(oppr𝑅))
5611, 47oppradd 19302 . . . 4 (+g𝑆) = (+g‘(oppr𝑆))
5724, 54, 55, 56isghm 18290 . . 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 235 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) GrpHom (oppr𝑆)))
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 19402 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3142  wf 6347  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Grpcgrp 18035   GrpHom cghm 18287  1rcur 19173  Ringcrg 19219  opprcoppr 19294   RingHom crh 19386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-grp 18038  df-ghm 18288  df-mgp 19162  df-ur 19174  df-ring 19221  df-oppr 19295  df-rnghom 19389
This theorem is referenced by:  elrhmunit  30809
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