Theorem elrhmunit 14314

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 2233	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
3		eqidd 2233	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
4		rhmrcl1 14292	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
5	4	adantr 276	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
6		ringsrg 14183	. . . . . 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 14245	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘𝑅))
10		eqid 2232	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2232	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	ringidcl 14156	. . . . 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 2233	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑅) = (1r‘𝑅))
15		eqidd 2233	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑅) = (∥r‘𝑅))
16		eqidd 2233	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑅) = (oppr‘𝑅))
17		eqidd 2233	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
18	3, 14, 15, 16, 17, 7	isunitd 14243	. . . . . 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 2232	. . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
22		eqid 2232	. . . . 5 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
23	10, 21, 22	rhmdvdsr 14312	. . . 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 1277	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)))
25		eqid 2232	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	11, 25	rhm1 14304	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
27	26	breq2d 4120	. . . 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 14313	. . . . 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 2232	. . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
33	32, 10	opprbasg 14211	. . . . . 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 2311	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘(oppr‘𝑅)))
36	13, 34	eleqtrd 2311	. . . 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 2232	. . . . 5 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
39		eqid 2232	. . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
40		eqid 2232	. . . . 5 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
41	38, 39, 40	rhmdvdsr 14312	. . . 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 1277	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)))
43	26	breq2d 4120	. . . 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 2233	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) = (Unit‘𝑆))
47		eqidd 2233	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑆) = (1r‘𝑆))
48		eqidd 2233	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑆) = (∥r‘𝑆))
49		eqidd 2233	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑆) = (oppr‘𝑆))
50		eqidd 2233	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
51		rhmrcl2 14293	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
52	51	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ Ring)
53		ringsrg 14183	. . . 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 14243	. 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 953	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203   class class class wbr 4108  ‘cfv 5351  (class class class)co 6049  Basecbs 13204  1_rcur 14095  SRingcsrg 14099  Ringcrg 14132  opp_rcoppr 14203  ∥_rcdsr 14222  Unitcui 14223   RingHom crh 14287

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-tpos 6475  df-map 6883  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-plusg 13295  df-mulr 13296  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-mhm 13664  df-grp 13708  df-minusg 13709  df-ghm 13950  df-cmn 13995  df-abl 13996  df-mgp 14057  df-ur 14096  df-srg 14100  df-ring 14134  df-oppr 14204  df-dvdsr 14225  df-unit 14226  df-rhm 14289

This theorem is referenced by:  rhmunitinv  14315