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 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

Proof of Theorem rhmopp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. 2 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
2		eqid 2732	. 2 ⊢ (1r‘(oppr‘𝑅)) = (1r‘(oppr‘𝑅))
3		eqid 2732	. 2 ⊢ (1r‘(oppr‘𝑆)) = (1r‘(oppr‘𝑆))
4		eqid 2732	. 2 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
5		eqid 2732	. 2 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
6		rhmrcl1 20247	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7		eqid 2732	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
8	7	opprringb 20154	. . 3 ⊢ (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring)
9	6, 8	sylib 217	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Ring)
10		rhmrcl2 20248	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11		eqid 2732	. . . 4 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
12	11	opprringb 20154	. . 3 ⊢ (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring)
13	10, 12	sylib 217	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Ring)
14		eqid 2732	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15	7, 14	oppr1 20156	. . . 4 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
16	15	eqcomi 2741	. . 3 ⊢ (1r‘(oppr‘𝑅)) = (1r‘𝑅)
17		eqid 2732	. . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆)
18	11, 17	oppr1 20156	. . . 4 ⊢ (1r‘𝑆) = (1r‘(oppr‘𝑆))
19	18	eqcomi 2741	. . 3 ⊢ (1r‘(oppr‘𝑆)) = (1r‘𝑆)
20	16, 19	rhm1 20259	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr‘𝑅))) = (1r‘(oppr‘𝑆)))
21		simpl 483	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22		simprr 771	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘(oppr‘𝑅)))
23		eqid 2732	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	7, 23	opprbas 20149	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
25	22, 24	eleqtrrdi 2844	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26		simprl 769	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘(oppr‘𝑅)))
27	26, 24	eleqtrrdi 2844	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28		eqid 2732	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
29		eqid 2732	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
30	23, 28, 29	rhmmul 20256	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(𝑦(.r‘𝑅)𝑥)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
31	21, 25, 27, 30	syl3anc 1371	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘(𝑦(.r‘𝑅)𝑥)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥)))
32	23, 28, 7, 4	opprmul 20145	. . . 4 ⊢ (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥)
33	32	fveq2i 6891	. . 3 ⊢ (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = (𝐹‘(𝑦(.r‘𝑅)𝑥))
34		eqid 2732	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
35	34, 29, 11, 5	opprmul 20145	. . 3 ⊢ ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥))
36	31, 33, 35	3eqtr4g 2797	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)))
37		ringgrp 20054	. . . . 5 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ Grp)
38	9, 37	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Grp)
39		ringgrp 20054	. . . . 5 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ Grp)
40	13, 39	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Grp)
41	23, 34	rhmf 20255	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42		rhmghm 20254	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
43	42	ad2antrr 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 ⊢ (+g‘𝑅) = (+g‘𝑅)
47		eqid 2732	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	23, 46, 47	ghmlin 19091	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
49	43, 44, 45, 48	syl3anc 1371	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
50	49	ralrimiva 3146	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
51	50	ralrimiva 3146	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
52	41, 51	jca 512	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))
53	38, 40, 52	jca31 515	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
54	11, 34	opprbas 20149	. . . 4 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
55	7, 46	oppradd 20151	. . . 4 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
56	11, 47	oppradd 20151	. . . 4 ⊢ (+g‘𝑆) = (+g‘(oppr‘𝑆))
57	24, 54, 55, 56	isghm 19086	. . 3 ⊢ (𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)) ↔ (((oppr‘𝑅) ∈ Grp ∧ (oppr‘𝑆) ∈ Grp) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
58	53, 57	sylibr 233	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)))
59	1, 2, 3, 4, 5, 9, 13, 20, 36, 58	isrhm2d 20257	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

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  1_rcur 19998  Ringcrg 20049  opp_rcoppr 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