Theorem rhmunitinv 14594

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

Assertion

Ref	Expression
rhmunitinv	|- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) )

Proof of Theorem rhmunitinv

Step	Hyp	Ref	Expression
1		rhmrcl1 14571	. . . . . 6 |- ( F e. ( R RingHom S ) -> R e. Ring )
2		eqid 2238	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
3		eqid 2238	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
4		eqid 2238	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
5		eqid 2238	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
6	2, 3, 4, 5	unitlinv 14542	. . . . . 6 |- ( ( R e. Ring /\ A e. (Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) )
7	1, 6	sylan 283	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) )
8	7	fveq2d 5702	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( F ` ( 1r ` R ) ) )
9		simpl 109	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( R RingHom S ) )
10		eqidd 2239	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
11		eqidd 2239	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
12	1	adantr 276	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. Ring )
13		ringsrg 14459	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
14	12, 13	syl 14	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. SRing )
15	10, 11, 14	unitssd 14525	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) C_ ( Base ` R ) )
16	2, 3	unitinvcl 14539	. . . . . . 7 |- ( ( R e. Ring /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. (Unit ` R ) )
17	1, 16	sylan 283	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. (Unit ` R ) )
18	15, 17	sseldd 3249	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. ( Base ` R ) )
19		simpr 110	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. (Unit ` R ) )
20	15, 19	sseldd 3249	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
21		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
22		eqid 2238	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
23	21, 4, 22	rhmmul 14580	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. ( Base ` R ) /\ A e. ( Base ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) )
24	9, 18, 20, 23	syl3anc 1278	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) )
25		eqid 2238	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	5, 25	rhm1 14583	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
28	8, 24, 27	3eqtr3d 2279	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
29		rhmrcl2 14572	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
30	29	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. Ring )
31		elrhmunit 14593	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
32		eqid 2238	. . . . 5 |- (Unit ` S ) = (Unit ` S )
33		eqid 2238	. . . . 5 |- ( invr ` S ) = ( invr ` S )
34	32, 33, 22, 25	unitlinv 14542	. . . 4 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
35	30, 31, 34	syl2anc 415	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
36	28, 35	eqtr4d 2274	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) )
37		eqidd 2239	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
38		eqid 2238	. . . . . . . 8 |- (mulGrp ` S ) = (mulGrp ` S )
39	38, 22	mgpplusgg 14328	. . . . . . 7 |- ( S e. Ring -> ( .r ` S ) = ( +g ` (mulGrp ` S ) ) )
40	30, 39	syl 14	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` (mulGrp ` S ) ) )
41		basfn 13469	. . . . . . . 8 |- Base Fn _V
42	30	elexd 2835	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. _V )
43		funfvex 5715	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
44	43	funfni 5486	. . . . . . . 8 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
45	41, 42, 44	sylancr 418	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) e. _V )
46		eqidd 2239	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) = ( Base ` S ) )
47		eqidd 2239	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
48		ringsrg 14459	. . . . . . . . 9 |- ( S e. Ring -> S e. SRing )
49	30, 48	syl 14	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. SRing )
50	46, 47, 49	unitssd 14525	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) C_ ( Base ` S ) )
51	45, 50	ssexd 4275	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) e. _V )
52	38	mgpex 14330	. . . . . . 7 |- ( S e. Ring -> (mulGrp ` S ) e. _V )
53	30, 52	syl 14	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (mulGrp ` S ) e. _V )
54	37, 40, 51, 53	ressplusgd 13542	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
55	54	oveqd 6105	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) )
56	54	oveqd 6105	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) )
57	55, 56	eqeq12d 2253	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) ) )
58		eqid 2238	. . . . . . 7 |- ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) )
59	32, 58	unitgrp 14532	. . . . . 6 |- ( S e. Ring -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
60	29, 59	syl 14	. . . . 5 |- ( F e. ( R RingHom S ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
61	60	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
62		elrhmunit 14593	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. (Unit ` S ) )
63	17, 62	syldan 282	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. (Unit ` S ) )
64	47, 37, 49	unitgrpbasd 14531	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
65	63, 64	eleqtrd 2317	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
66	32, 33	unitinvcl 14539	. . . . . 6 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
67	30, 31, 66	syl2anc 415	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
68	67, 64	eleqtrd 2317	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
69	31, 64	eleqtrd 2317	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
70		eqid 2238	. . . . 5 |- ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
71		eqid 2238	. . . . 5 |- ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
72	70, 71	grprcan 13901	. . . 4 |- ( ( ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp /\ ( ( F ` ( ( invr ` R ) ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) /\ ( ( invr ` S ) ` ( F ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) /\ ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
73	61, 65, 68, 69, 72	syl13anc 1280	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
74	57, 73	bitrd 188	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
75	36, 74	mpbid 147	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   Basecbs 13410   ↾_s cress 13411   +g cplusg 13490   .rcmulr 13491   Grpcgrp 13864  mulGrpcmgp 14324   1rcur 14369  SRingcsrg 14374   Ringcrg 14407  Unitcui 14502   invrcinvr 14536   RingHom crh 14566

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 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

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 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-mhm 13825  df-grp 13867  df-minusg 13868  df-ghm 14103  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-oppr 14482  df-dvdsr 14504  df-unit 14505  df-invr 14537  df-rhm 14568

This theorem is used by: (None)