Theorem elrhmunit 14594

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 2239	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
3		eqidd 2239	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
4		rhmrcl1 14572	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
5	4	adantr 276	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
6		ringsrg 14460	. . . . . 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 14525	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘𝑅))
10		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2238	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	ringidcl 14433	. . . . 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 2239	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑅) = (1r‘𝑅))
15		eqidd 2239	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑅) = (∥r‘𝑅))
16		eqidd 2239	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑅) = (oppr‘𝑅))
17		eqidd 2239	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
18	3, 14, 15, 16, 17, 7	isunitd 14523	. . . . . 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 2238	. . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
22		eqid 2238	. . . . 5 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
23	10, 21, 22	rhmdvdsr 14592	. . . 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 1281	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘𝑆)(𝐹‘(1r‘𝑅)))
25		eqid 2238	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	11, 25	rhm1 14584	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
27	26	breq2d 4143	. . . 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 14593	. . . . 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 2238	. . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
33	32, 10	opprbasg 14490	. . . . . 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 2317	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘(oppr‘𝑅)))
36	13, 34	eleqtrd 2317	. . . 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 2238	. . . . 5 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
39		eqid 2238	. . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
40		eqid 2238	. . . . 5 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
41	38, 39, 40	rhmdvdsr 14592	. . . 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 1281	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴)(∥r‘(oppr‘𝑆))(𝐹‘(1r‘𝑅)))
43	26	breq2d 4143	. . . 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 2239	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) = (Unit‘𝑆))
47		eqidd 2239	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (1r‘𝑆) = (1r‘𝑆))
48		eqidd 2239	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘𝑆) = (∥r‘𝑆))
49		eqidd 2239	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (oppr‘𝑆) = (oppr‘𝑆))
50		eqidd 2239	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
51		rhmrcl2 14573	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
52	51	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ Ring)
53		ringsrg 14460	. . . 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 14523	. 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 957	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  Basecbs 13411  1_rcur 14370  SRingcsrg 14375  Ringcrg 14408  opp_rcoppr 14482  ∥_rcdsr 14502  Unitcui 14503   RingHom crh 14567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-pre-ltirr 8298  ax-pre-lttrn 8300  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6520  df-map 6928  df-pnf 8369  df-mnf 8370  df-ltxr 8372  df-inn 9315  df-2 9373  df-3 9374  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-plusg 13504  df-mulr 13505  df-0g 13672  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-mhm 13826  df-grp 13868  df-minusg 13869  df-ghm 14104  df-cmn 14180  df-abl 14181  df-mgp 14326  df-ur 14371  df-srg 14376  df-ring 14410  df-oppr 14483  df-dvdsr 14505  df-unit 14506  df-rhm 14569

This theorem is used by:  rhmunitinv  14595