Theorem elrhmunit 14184

Description: Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)

Assertion

Ref	Expression
elrhmunit	⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))

Proof of Theorem elrhmunit

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
2		eqidd 2230	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
3		eqidd 2230	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
4		rhmrcl1 14162	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
5	4	adantr 276	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
6		ringsrg 14053	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
7	5, 6	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ SRing)
8		simpr 110	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Unit‘𝑅))
9	2, 3, 7, 8	unitcld 14115	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘𝑅))
10		eqid 2229	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2229	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	ringidcl 14026	. . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
13	1, 4, 12	3syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
14		eqidd 2230	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑅) = (1r‘𝑅))
15		eqidd 2230	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑅) = (∥r‘𝑅))
16		eqidd 2230	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑅) = (oppr‘𝑅))
17		eqidd 2230	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
18	3, 14, 15, 16, 17, 7	isunitd 14113	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐴 ∈ (Unit‘𝑅) ↔ (𝐴(∥r‘𝑅)(1r‘𝑅) ∧ 𝐴(∥r‘(oppr‘𝑅))(1r‘𝑅))))
19	8, 18	mpbid 147	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐴(∥r‘𝑅)(1r‘𝑅) ∧ 𝐴(∥r‘(oppr‘𝑅))(1r‘𝑅)))
20	19	simpld 112	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴(∥r‘𝑅)(1r‘𝑅))
21		eqid 2229	. . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
22		eqid 2229	. . . . 5 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
23	10, 21, 22	rhmdvdsr 14182	. . . 4 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Base‘𝑅) ∧ (1r‘𝑅) ∈ (Base‘𝑅)) ∧ 𝐴(∥r‘𝑅)(1r‘𝑅)) → (𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)))
24	1, 9, 13, 20, 23	syl31anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)))
25		eqid 2229	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	11, 25	rhm1 14174	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
27	26	breq2d 4098	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)) ↔ (𝐹‘𝐴)(∥r‘𝑆)(1r‘𝑆)))
28	27	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)) ↔ (𝐹‘𝐴)(∥r‘𝑆)(1r‘𝑆)))
29	24, 28	mpbid 147	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘𝑆)(1r‘𝑆))
30		rhmopp 14183	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))
31	30	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))
32		eqid 2229	. . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
33	32, 10	opprbasg 14081	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
34	5, 33	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
35	9, 34	eleqtrd 2308	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘(oppr‘𝑅)))
36	13, 34	eleqtrd 2308	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑅) ∈ (Base‘(oppr‘𝑅)))
37	19	simprd 114	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴(∥r‘(oppr‘𝑅))(1r‘𝑅))
38		eqid 2229	. . . . 5 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
39		eqid 2229	. . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
40		eqid 2229	. . . . 5 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
41	38, 39, 40	rhmdvdsr 14182	. . . 4 ⊢ (((𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)) ∧ 𝐴 ∈ (Base‘(oppr‘𝑅)) ∧ (1r‘𝑅) ∈ (Base‘(oppr‘𝑅))) ∧ 𝐴(∥r‘(oppr‘𝑅))(1r‘𝑅)) → (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)))
42	31, 35, 36, 37, 41	syl31anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)))
43	26	breq2d 4098	. . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)) ↔ (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(1r‘𝑆)))
44	43	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)) ↔ (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(1r‘𝑆)))
45	42, 44	mpbid 147	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(1r‘𝑆))
46		eqidd 2230	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) = (Unit‘𝑆))
47		eqidd 2230	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑆) = (1r‘𝑆))
48		eqidd 2230	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑆) = (∥r‘𝑆))
49		eqidd 2230	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑆) = (oppr‘𝑆))
50		eqidd 2230	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
51		rhmrcl2 14163	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
52	51	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ Ring)
53		ringsrg 14053	. . . 4 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
54	52, 53	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ SRing)
55	46, 47, 48, 49, 50, 54	isunitd 14113	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘𝐴) ∈ (Unit‘𝑆) ↔ ((𝐹‘𝐴)(∥r‘𝑆)(1r‘𝑆) ∧ (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(1r‘𝑆))))
56	29, 45, 55	mpbir2and 950	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  Basecbs 13075  1_rcur 13965  SRingcsrg 13969  Ringcrg 14002  opp_rcoppr 14073  ∥_rcdsr 14092  Unitcui 14093   RingHom crh 14157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-tpos 6406  df-map 6814  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-plusg 13166  df-mulr 13167  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-mhm 13535  df-grp 13579  df-minusg 13580  df-ghm 13821  df-cmn 13866  df-abl 13867  df-mgp 13927  df-ur 13966  df-srg 13970  df-ring 14004  df-oppr 14074  df-dvdsr 14095  df-unit 14096  df-rhm 14159

This theorem is referenced by:  rhmunitinv  14185