Theorem rhmunitinv 14423

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 14400	. . . . . 6 |- ( F e. ( R RingHom S ) -> R e. Ring )
2		eqid 2234	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
3		eqid 2234	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
4		eqid 2234	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
5		eqid 2234	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
6	2, 3, 4, 5	unitlinv 14371	. . . . . 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 5679	. . . 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 2235	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
11		eqidd 2235	. . . . . . 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 14290	. . . . . . . 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 14354	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) C_ ( Base ` R ) )
16	2, 3	unitinvcl 14368	. . . . . . 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 3243	. . . . 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 3243	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
21		eqid 2234	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
22		eqid 2234	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
23	21, 4, 22	rhmmul 14409	. . . . 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 1274	. . . 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 2234	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	5, 25	rhm1 14412	. . . . 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 2275	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
29		rhmrcl2 14401	. . . . 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 14422	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
32		eqid 2234	. . . . 5 |- (Unit ` S ) = (Unit ` S )
33		eqid 2234	. . . . 5 |- ( invr ` S ) = ( invr ` S )
34	32, 33, 22, 25	unitlinv 14371	. . . 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 2270	. 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 2235	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
38		eqid 2234	. . . . . . . 8 |- (mulGrp ` S ) = (mulGrp ` S )
39	38, 22	mgpplusgg 14163	. . . . . . 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 13355	. . . . . . . 8 |- Base Fn _V
42	30	elexd 2829	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. _V )
43		funfvex 5692	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
44	43	funfni 5463	. . . . . . . 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 2235	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) = ( Base ` S ) )
47		eqidd 2235	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
48		ringsrg 14290	. . . . . . . . 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 14354	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) C_ ( Base ` S ) )
51	45, 50	ssexd 4255	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) e. _V )
52	38	mgpex 14164	. . . . . . 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 13426	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
55	54	oveqd 6075	. . . 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 6075	. . . 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 2249	. . 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 2234	. . . . . . 7 |- ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) )
59	32, 58	unitgrp 14361	. . . . . 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 14422	. . . . . 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 14360	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
65	63, 64	eleqtrd 2313	. . . 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 14368	. . . . . 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 2313	. . . 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 2313	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
70		eqid 2234	. . . . 5 |- ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
71		eqid 2234	. . . . 5 |- ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
72	70, 71	grprcan 13792	. . . 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 1276	. . 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 1398   e. wcel 2205   _Vcvv 2815   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   .rcmulr 13375   Grpcgrp 13755  mulGrpcmgp 14159   1rcur 14202  SRingcsrg 14206   Ringcrg 14239  Unitcui 14331   invrcinvr 14365   RingHom crh 14395

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-minusg 13759  df-ghm 13994  df-cmn 14039  df-abl 14040  df-mgp 14160  df-ur 14203  df-srg 14207  df-ring 14241  df-oppr 14311  df-dvdsr 14333  df-unit 14334  df-invr 14366  df-rhm 14397

This theorem is referenced by: (None)