Theorem rhmopp 20792

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 2763	. 2 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
2		eqid 2763	. 2 ⊢ (1r‘(oppr‘𝑅)) = (1r‘(oppr‘𝑅))
3		eqid 2763	. 2 ⊢ (1r‘(oppr‘𝑆)) = (1r‘(oppr‘𝑆))
4		eqid 2763	. 2 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
5		eqid 2763	. 2 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
6		rhmrcl1 20740	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
7		eqid 2763	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
8	7	opprringb 20611	. . 3 ⊢ (𝑅 ∈ Ring ↔ (oppr‘𝑅) ∈ Ring)
9	6, 8	sylib 221	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Ring)
10		rhmrcl2 20741	. . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
11		eqid 2763	. . . 4 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
12	11	opprringb 20611	. . 3 ⊢ (𝑆 ∈ Ring ↔ (oppr‘𝑆) ∈ Ring)
13	10, 12	sylib 221	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Ring)
14		eqid 2763	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15	7, 14	oppr1 20613	. . . 4 ⊢ (1r‘𝑅) = (1r‘(oppr‘𝑅))
16	15	eqcomi 2772	. . 3 ⊢ (1r‘(oppr‘𝑅)) = (1r‘𝑅)
17		eqid 2763	. . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆)
18	11, 17	oppr1 20613	. . . 4 ⊢ (1r‘𝑆) = (1r‘(oppr‘𝑆))
19	18	eqcomi 2772	. . 3 ⊢ (1r‘(oppr‘𝑆)) = (1r‘𝑆)
20	16, 19	rhm1 20757	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘(oppr‘𝑅))) = (1r‘(oppr‘𝑆)))
21		simpl 488	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝐹 ∈ (𝑅 RingHom 𝑆))
22		simprr 785	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘(oppr‘𝑅)))
23		eqid 2763	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	7, 23	opprbas 20606	. . . . 5 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
25	22, 24	eleqtrrdi 2875	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
26		simprl 783	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘(oppr‘𝑅)))
27	26, 24	eleqtrrdi 2875	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑥 ∈ (Base‘(oppr‘𝑅)) ∧ 𝑦 ∈ (Base‘(oppr‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
28		eqid 2763	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
29		eqid 2763	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
30	23, 28, 29	rhmmul 20753	. . . 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 1399	. . 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 20603	. . . 4 ⊢ (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥)
33	32	fveq2i 6914	. . 3 ⊢ (𝐹‘(𝑥(.r‘(oppr‘𝑅))𝑦)) = (𝐹‘(𝑦(.r‘𝑅)𝑥))
34		eqid 2763	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
35	34, 29, 11, 5	opprmul 20603	. . 3 ⊢ ((𝐹‘𝑥)(.r‘(oppr‘𝑆))(𝐹‘𝑦)) = ((𝐹‘𝑦)(.r‘𝑆)(𝐹‘𝑥))
36	31, 33, 35	3eqtr4g 2823	. 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 20497	. . . . 5 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ Grp)
38	9, 37	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑅) ∈ Grp)
39		ringgrp 20497	. . . . 5 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ Grp)
40	13, 39	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (oppr‘𝑆) ∈ Grp)
41	23, 34	rhmf 20748	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
42		rhmghm 20747	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
43	42	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
44		simplr 781	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
45		simpr 490	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
46		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
47		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
48	23, 46, 47	ghmlin 19467	. . . . . . . 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 1399	. . . . . . 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 3158	. . . . . 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 3158	. . . . 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 521	. . . 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 524	. . 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 20606	. . . 4 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
55	7, 46	oppradd 20607	. . . 4 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅))
56	11, 47	oppradd 20607	. . . 4 ⊢ (+g‘𝑆) = (+g‘(oppr‘𝑆))
57	24, 54, 55, 56	isghm 19462	. . 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 237	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) GrpHom (oppr‘𝑆)))
59	1, 2, 3, 4, 5, 9, 13, 20, 36, 58	isrhm2d 20754	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  +_gcplusg 17460  ._rcmulr 17461  Grpcgrp 19176   GrpHom cghm 19459  1_rcur 20440  Ringcrg 20492  opp_rcoppr 20599   RingHom crh 20732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-tpos 8274  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-mulr 17474  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mhm 19010  df-grp 19179  df-minusg 19180  df-ghm 19460  df-cmn 20029  df-abl 20030  df-mgp 20394  df-rng 20408  df-ur 20441  df-ring 20494  df-oppr 20600  df-rhm 20735

This theorem is used by:  elrhmunit  20793  ker2idl  21606