Theorem rhmopp 20486

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 2736	. 2 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
2		eqid 2736	. 2 ⊢ (1r‘(oppr‘𝑅)) = (1r‘(oppr‘𝑅))
3		eqid 2736	. 2 ⊢ (1r‘(oppr‘𝑆)) = (1r‘(oppr‘𝑆))
4		eqid 2736	. 2 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
5		eqid 2736	. 2 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
6		rhmrcl1 20456	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7		eqid 2736	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
8	7	opprringb 20328	. . 3 ⊢ (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring)
9	6, 8	sylib 218	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Ring)
10		rhmrcl2 20457	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11		eqid 2736	. . . 4 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
12	11	opprringb 20328	. . 3 ⊢ (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring)
13	10, 12	sylib 218	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Ring)
14		eqid 2736	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15	7, 14	oppr1 20330	. . . 4 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
16	15	eqcomi 2745	. . 3 ⊢ (1r‘(oppr‘𝑅)) = (1r‘𝑅)
17		eqid 2736	. . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆)
18	11, 17	oppr1 20330	. . . 4 ⊢ (1r‘𝑆) = (1r‘(oppr‘𝑆))
19	18	eqcomi 2745	. . 3 ⊢ (1r‘(oppr‘𝑆)) = (1r‘𝑆)
20	16, 19	rhm1 20468	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr‘𝑅))) = (1r‘(oppr‘𝑆)))
21		simpl 482	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22		simprr 773	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘(oppr‘𝑅)))
23		eqid 2736	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	7, 23	opprbas 20323	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
25	22, 24	eleqtrrdi 2847	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26		simprl 771	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘(oppr‘𝑅)))
27	26, 24	eleqtrrdi 2847	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28		eqid 2736	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
29		eqid 2736	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
30	23, 28, 29	rhmmul 20465	. . . 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 1374	. . 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 20320	. . . 4 ⊢ (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥)
33	32	fveq2i 6843	. . 3 ⊢ (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = (𝐹‘(𝑦(.r‘𝑅)𝑥))
34		eqid 2736	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
35	34, 29, 11, 5	opprmul 20320	. . 3 ⊢ ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥))
36	31, 33, 35	3eqtr4g 2796	. 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 20219	. . . . 5 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ Grp)
38	9, 37	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Grp)
39		ringgrp 20219	. . . . 5 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ Grp)
40	13, 39	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Grp)
41	23, 34	rhmf 20464	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42		rhmghm 20463	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
43	42	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44		simplr 769	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
47		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	23, 46, 47	ghmlin 19196	. . . . . . . 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 1374	. . . . . . 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 3129	. . . . . 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 3129	. . . . 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 511	. . . 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 514	. . 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 20323	. . . 4 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
55	7, 46	oppradd 20324	. . . 4 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
56	11, 47	oppradd 20324	. . . 4 ⊢ (+g‘𝑆) = (+g‘(oppr‘𝑆))
57	24, 54, 55, 56	isghm 19190	. . 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 234	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)))
59	1, 2, 3, 4, 5, 9, 13, 20, 36, 58	isrhm2d 20466	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Grpcgrp 18909   GrpHom cghm 19187  1_rcur 20162  Ringcrg 20214  opp_rcoppr 20316   RingHom crh 20449

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-grp 18912  df-minusg 18913  df-ghm 19188  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-oppr 20317  df-rhm 20452

This theorem is referenced by:  elrhmunit  20487