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Theorem rhmopp 20280
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 2732 . 2 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
2 eqid 2732 . 2 (1r‘(oppr𝑅)) = (1r‘(oppr𝑅))
3 eqid 2732 . 2 (1r‘(oppr𝑆)) = (1r‘(oppr𝑆))
4 eqid 2732 . 2 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
5 eqid 2732 . 2 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
6 rhmrcl1 20247 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7 eqid 2732 . . . 4 (oppr𝑅) = (oppr𝑅)
87opprringb 20154 . . 3 (𝑅 ∈ Ring ↔ (oppr𝑅) ∈ Ring)
96, 8sylib 217 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Ring)
10 rhmrcl2 20248 . . 3 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11 eqid 2732 . . . 4 (oppr𝑆) = (oppr𝑆)
1211opprringb 20154 . . 3 (𝑆 ∈ Ring ↔ (oppr𝑆) ∈ Ring)
1310, 12sylib 217 . 2 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Ring)
14 eqid 2732 . . . . 5 (1r𝑅) = (1r𝑅)
157, 14oppr1 20156 . . . 4 (1r𝑅) = (1r‘(oppr𝑅))
1615eqcomi 2741 . . 3 (1r‘(oppr𝑅)) = (1r𝑅)
17 eqid 2732 . . . . 5 (1r𝑆) = (1r𝑆)
1811, 17oppr1 20156 . . . 4 (1r𝑆) = (1r‘(oppr𝑆))
1918eqcomi 2741 . . 3 (1r‘(oppr𝑆)) = (1r𝑆)
2016, 19rhm1 20259 . 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 771 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘(oppr𝑅)))
23 eqid 2732 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
247, 23opprbas 20149 . . . . 5 (Base‘𝑅) = (Base‘(oppr𝑅))
2522, 24eleqtrrdi 2844 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26 simprl 769 . . . . 5 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘(oppr𝑅)))
2726, 24eleqtrrdi 2844 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28 eqid 2732 . . . . 5 (.r𝑅) = (.r𝑅)
29 eqid 2732 . . . . 5 (.r𝑆) = (.r𝑆)
3023, 28, 29rhmmul 20256 . . . 4 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3121, 25, 27, 30syl3anc 1371 . . 3 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑦(.r𝑅)𝑥)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥)))
3223, 28, 7, 4opprmul 20145 . . . 4 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
3332fveq2i 6891 . . 3 (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = (𝐹‘(𝑦(.r𝑅)𝑥))
34 eqid 2732 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3534, 29, 11, 5opprmul 20145 . . 3 ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)) = ((𝐹𝑦)(.r𝑆)(𝐹𝑥))
3631, 33, 353eqtr4g 2797 . 2 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr𝑅)) ∧ 𝑦 ∈ (Base‘(oppr𝑅)))) → (𝐹‘(𝑥(.r‘(oppr𝑅))𝑦)) = ((𝐹𝑥)(.r‘(oppr𝑆))(𝐹𝑦)))
37 ringgrp 20054 . . . . 5 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ Grp)
389, 37syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑅) ∈ Grp)
39 ringgrp 20054 . . . . 5 ((oppr𝑆) ∈ Ring → (oppr𝑆) ∈ Grp)
4013, 39syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr𝑆) ∈ Grp)
4123, 34rhmf 20255 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42 rhmghm 20254 . . . . . . . . 9 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4342ad2antrr 724 . . . . . . . 8 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44 simplr 767 . . . . . . . 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 2732 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
47 eqid 2732 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
4823, 46, 47ghmlin 19091 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
4943, 44, 45, 48syl3anc 1371 . . . . . . 7 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5049ralrimiva 3146 . . . . . 6 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
5150ralrimiva 3146 . . . . 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 20149 . . . 4 (Base‘𝑆) = (Base‘(oppr𝑆))
557, 46oppradd 20151 . . . 4 (+g𝑅) = (+g‘(oppr𝑅))
5611, 47oppradd 20151 . . . 4 (+g𝑆) = (+g‘(oppr𝑆))
5724, 54, 55, 56isghm 19086 . . 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 20257 1 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  wf 6536  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  Grpcgrp 18815   GrpHom cghm 19083  1rcur 19998  Ringcrg 20049  opprcoppr 20141   RingHom crh 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-ghm 19084  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-rnghom 20243
This theorem is referenced by:  elrhmunit  20281
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