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Theorem rhmopp 20448
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 2728 . 2 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
2 eqid 2728 . 2 (1r‘(oppr𝑅)) = (1r‘(oppr𝑅))
3 eqid 2728 . 2 (1r‘(oppr𝑆)) = (1r‘(oppr𝑆))
4 eqid 2728 . 2 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
5 eqid 2728 . 2 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
6 rhmrcl1 20415 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7 eqid 2728 . . . 4 (oppr𝑅) = (oppr𝑅)
87opprringb 20287 . . 3 (𝑅 ∈ Ring ↔ (oppr𝑅) ∈ Ring)
96, 8sylib 217 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Ring)
10 rhmrcl2 20416 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11 eqid 2728 . . . 4 (oppr𝑆) = (oppr𝑆)
1211opprringb 20287 . . 3 (𝑆 ∈ Ring ↔ (oppr𝑆) ∈ Ring)
1310, 12sylib 217 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Ring)
14 eqid 2728 . . . . 5 (1r𝑅) = (1r𝑅)
157, 14oppr1 20289 . . . 4 (1r𝑅) = (1r‘(oppr𝑅))
1615eqcomi 2737 . . 3 (1r‘(oppr𝑅)) = (1r𝑅)
17 eqid 2728 . . . . 5 (1r𝑆) = (1r𝑆)
1811, 17oppr1 20289 . . . 4 (1r𝑆) = (1r‘(oppr𝑆))
1918eqcomi 2737 . . 3 (1r‘(oppr𝑆)) = (1r𝑆)
2016, 19rhm1 20428 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr𝑅))) = (1r‘(oppr𝑆)))
21 simpl 482 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22 simprr 772 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘(oppr𝑅)))
23 eqid 2728 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
247, 23opprbas 20280 . . . . 5 (Base‘𝑅) = (Base‘(oppr𝑅))
2522, 24eleqtrrdi 2840 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26 simprl 770 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘(oppr𝑅)))
2726, 24eleqtrrdi 2840 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28 eqid 2728 . . . . 5 (.r𝑅) = (.r𝑅)
29 eqid 2728 . . . . 5 (.r𝑆) = (.r𝑆)
3023, 28, 29rhmmul 20425 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3121, 25, 27, 30syl3anc 1369 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3223, 28, 7, 4opprmul 20276 . . . 4 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
3332fveq2i 6900 . . 3 (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = (𝐹‘(𝑦(.r𝑅)𝑥))
34 eqid 2728 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3534, 29, 11, 5opprmul 20276 . . 3 ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥))
3631, 33, 353eqtr4g 2793 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)))
37 ringgrp 20178 . . . . 5 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ Grp)
389, 37syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Grp)
39 ringgrp 20178 . . . . 5 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ Grp)
4013, 39syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Grp)
4123, 34rhmf 20424 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42 rhmghm 20423 . . . . . . . . 9 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4342ad2antrr 725 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44 simplr 768 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
45 simpr 484 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
46 eqid 2728 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
47 eqid 2728 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
4823, 46, 47ghmlin 19175 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
4943, 44, 45, 48syl3anc 1369 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5049ralrimiva 3143 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5150ralrimiva 3143 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5241, 51jca 511 . . . 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 514 . . 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 20280 . . . 4 (Base‘𝑆) = (Base‘(oppr𝑆))
557, 46oppradd 20282 . . . 4 (+g𝑅) = (+g‘(oppr𝑅))
5611, 47oppradd 20282 . . . 4 (+g𝑆) = (+g‘(oppr𝑆))
5724, 54, 55, 56isghm 19170 . . 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 233 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) GrpHom (oppr𝑆)))
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 20426 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  wf 6544  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  .rcmulr 17234  Grpcgrp 18890   GrpHom cghm 19167  1rcur 20121  Ringcrg 20173  opprcoppr 20272   RingHom crh 20408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-plusg 17246  df-mulr 17247  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18740  df-grp 18893  df-minusg 18894  df-ghm 19168  df-cmn 19737  df-abl 19738  df-mgp 20075  df-rng 20093  df-ur 20122  df-ring 20175  df-oppr 20273  df-rhm 20411
This theorem is referenced by:  elrhmunit  20449
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