Theorem rhmopp 20409

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 2726	. 2 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
2		eqid 2726	. 2 ⊢ (1r‘(oppr‘𝑅)) = (1r‘(oppr‘𝑅))
3		eqid 2726	. 2 ⊢ (1r‘(oppr‘𝑆)) = (1r‘(oppr‘𝑆))
4		eqid 2726	. 2 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
5		eqid 2726	. 2 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
6		rhmrcl1 20376	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7		eqid 2726	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
8	7	opprringb 20248	. . 3 ⊢ (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring)
9	6, 8	sylib 217	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Ring)
10		rhmrcl2 20377	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11		eqid 2726	. . . 4 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
12	11	opprringb 20248	. . 3 ⊢ (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring)
13	10, 12	sylib 217	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Ring)
14		eqid 2726	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15	7, 14	oppr1 20250	. . . 4 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
16	15	eqcomi 2735	. . 3 ⊢ (1r‘(oppr‘𝑅)) = (1r‘𝑅)
17		eqid 2726	. . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆)
18	11, 17	oppr1 20250	. . . 4 ⊢ (1r‘𝑆) = (1r‘(oppr‘𝑆))
19	18	eqcomi 2735	. . 3 ⊢ (1r‘(oppr‘𝑆)) = (1r‘𝑆)
20	16, 19	rhm1 20389	. 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 770	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘(oppr‘𝑅)))
23		eqid 2726	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	7, 23	opprbas 20241	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
25	22, 24	eleqtrrdi 2838	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26		simprl 768	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘(oppr‘𝑅)))
27	26, 24	eleqtrrdi 2838	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28		eqid 2726	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
29		eqid 2726	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
30	23, 28, 29	rhmmul 20386	. . . 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 1368	. . 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 20237	. . . 4 ⊢ (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥)
33	32	fveq2i 6887	. . 3 ⊢ (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = (𝐹‘(𝑦(.r‘𝑅)𝑥))
34		eqid 2726	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
35	34, 29, 11, 5	opprmul 20237	. . 3 ⊢ ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥))
36	31, 33, 35	3eqtr4g 2791	. 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 20141	. . . . 5 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ Grp)
38	9, 37	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Grp)
39		ringgrp 20141	. . . . 5 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ Grp)
40	13, 39	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Grp)
41	23, 34	rhmf 20385	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42		rhmghm 20384	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
43	42	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44		simplr 766	. . . . . . . 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 2726	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
47		eqid 2726	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	23, 46, 47	ghmlin 19144	. . . . . . . 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 1368	. . . . . . 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 3140	. . . . . 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 3140	. . . . 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 20241	. . . 4 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
55	7, 46	oppradd 20243	. . . 4 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
56	11, 47	oppradd 20243	. . . 4 ⊢ (+g‘𝑆) = (+g‘(oppr‘𝑆))
57	24, 54, 55, 56	isghm 19139	. . 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 20387	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  +_gcplusg 17204  ._rcmulr 17205  Grpcgrp 18861   GrpHom cghm 19136  1_rcur 20084  Ringcrg 20136  opp_rcoppr 20233   RingHom crh 20369

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-tpos 8209  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-mulr 17218  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-grp 18864  df-minusg 18865  df-ghm 19137  df-cmn 19700  df-abl 19701  df-mgp 20038  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20234  df-rhm 20372

This theorem is referenced by:  elrhmunit  20410