Theorem rhmunitinv 14211

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 14188	. . . . . 6 |- ( F e. ( R RingHom S ) -> R e. Ring )
2		eqid 2231	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
3		eqid 2231	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
4		eqid 2231	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
5		eqid 2231	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
6	2, 3, 4, 5	unitlinv 14159	. . . . . 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 5643	. . . 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 2232	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
11		eqidd 2232	. . . . . . 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 14079	. . . . . . . 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 14142	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) C_ ( Base ` R ) )
16	2, 3	unitinvcl 14156	. . . . . . 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 3228	. . . . 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 3228	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
21		eqid 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
22		eqid 2231	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
23	21, 4, 22	rhmmul 14197	. . . . 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 1273	. . . 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 2231	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	5, 25	rhm1 14200	. . . . 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 2272	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
29		rhmrcl2 14189	. . . . 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 14210	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
32		eqid 2231	. . . . 5 |- (Unit ` S ) = (Unit ` S )
33		eqid 2231	. . . . 5 |- ( invr ` S ) = ( invr ` S )
34	32, 33, 22, 25	unitlinv 14159	. . . 4 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
35	30, 31, 34	syl2anc 411	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
36	28, 35	eqtr4d 2267	. 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 2232	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
38		eqid 2231	. . . . . . . 8 |- (mulGrp ` S ) = (mulGrp ` S )
39	38, 22	mgpplusgg 13956	. . . . . . 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 13159	. . . . . . . 8 |- Base Fn _V
42	30	elexd 2816	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. _V )
43		funfvex 5656	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
44	43	funfni 5432	. . . . . . . 8 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
45	41, 42, 44	sylancr 414	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) e. _V )
46		eqidd 2232	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) = ( Base ` S ) )
47		eqidd 2232	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
48		ringsrg 14079	. . . . . . . . 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 14142	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) C_ ( Base ` S ) )
51	45, 50	ssexd 4229	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) e. _V )
52	38	mgpex 13957	. . . . . . 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 13230	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
55	54	oveqd 6035	. . . 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 6035	. . . 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 2246	. . 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 2231	. . . . . . 7 |- ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) )
59	32, 58	unitgrp 14149	. . . . . 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 14210	. . . . . 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 14148	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
65	63, 64	eleqtrd 2310	. . . 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 14156	. . . . . 6 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
67	30, 31, 66	syl2anc 411	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
68	67, 64	eleqtrd 2310	. . . 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 2310	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
70		eqid 2231	. . . . 5 |- ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
71		eqid 2231	. . . . 5 |- ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
72	70, 71	grprcan 13638	. . . 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 1275	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13100   ↾_s cress 13101   +g cplusg 13178   .rcmulr 13179   Grpcgrp 13601  mulGrpcmgp 13952   1rcur 13991  SRingcsrg 13995   Ringcrg 14028  Unitcui 14119   invrcinvr 14153   RingHom crh 14183

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-tpos 6411  df-map 6819  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-mhm 13560  df-grp 13604  df-minusg 13605  df-ghm 13846  df-cmn 13891  df-abl 13892  df-mgp 13953  df-ur 13992  df-srg 13996  df-ring 14030  df-oppr 14100  df-dvdsr 14121  df-unit 14122  df-invr 14154  df-rhm 14185

This theorem is referenced by: (None)