Theorem lmhmf1o 19046

Description: A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypotheses

Ref	Expression
lmhmf1o.x	|- X = ( Base ` S )
lmhmf1o.y	|- Y = ( Base ` T )

Assertion

Ref	Expression
lmhmf1o	|- ( F e. ( S LMHom T ) -> ( F : X -1-1-onto-> Y <-> `' F e. ( T LMHom S ) ) )

Proof of Theorem lmhmf1o

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmf1o.y	. . 3 |- Y = ( Base ` T )
2		eqid 2622	. . 3 |- ( .s ` T ) = ( .s ` T )
3		eqid 2622	. . 3 |- ( .s ` S ) = ( .s ` S )
4		eqid 2622	. . 3 |- (Scalar ` T ) = (Scalar ` T )
5		eqid 2622	. . 3 |- (Scalar ` S ) = (Scalar ` S )
6		eqid 2622	. . 3 |- ( Base ` (Scalar ` T ) ) = ( Base ` (Scalar ` T ) )
7		lmhmlmod2 19032	. . . 4 |- ( F e. ( S LMHom T ) -> T e. LMod )
8	7	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> T e. LMod )
9		lmhmlmod1 19033	. . . 4 |- ( F e. ( S LMHom T ) -> S e. LMod )
10	9	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> S e. LMod )
11	5, 4	lmhmsca 19030	. . . . 5 |- ( F e. ( S LMHom T ) -> (Scalar ` T ) = (Scalar ` S ) )
12	11	eqcomd 2628	. . . 4 |- ( F e. ( S LMHom T ) -> (Scalar ` S ) = (Scalar ` T ) )
13	12	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> (Scalar ` S ) = (Scalar ` T ) )
14		lmghm 19031	. . . . 5 |- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) )
15		lmhmf1o.x	. . . . . 6 |- X = ( Base ` S )
16	15, 1	ghmf1o 17690	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( F : X -1-1-onto-> Y <-> `' F e. ( T GrpHom S ) ) )
17	14, 16	syl 17	. . . 4 |- ( F e. ( S LMHom T ) -> ( F : X -1-1-onto-> Y <-> `' F e. ( T GrpHom S ) ) )
18	17	biimpa 501	. . 3 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> `' F e. ( T GrpHom S ) )
19		simpll 790	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> F e. ( S LMHom T ) )
20	13	fveq2d 6195	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` T ) ) )
21	20	eleq2d 2687	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> ( a e. ( Base ` (Scalar ` S ) ) <-> a e. ( Base ` (Scalar ` T ) ) ) )
22	21	biimpar 502	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ a e. ( Base ` (Scalar ` T ) ) ) -> a e. ( Base ` (Scalar ` S ) ) )
23	22	adantrr 753	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> a e. ( Base ` (Scalar ` S ) ) )
24		f1ocnv 6149	. . . . . . . . . 10 |- ( F : X -1-1-onto-> Y -> `' F : Y -1-1-onto-> X )
25		f1of 6137	. . . . . . . . . 10 |- ( `' F : Y -1-1-onto-> X -> `' F : Y --> X )
26	24, 25	syl 17	. . . . . . . . 9 |- ( F : X -1-1-onto-> Y -> `' F : Y --> X )
27	26	adantl 482	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> `' F : Y --> X )
28	27	ffvelrnda 6359	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ b e. Y ) -> ( `' F ` b ) e. X )
29	28	adantrl 752	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( `' F ` b ) e. X )
30		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` S ) )
31	5, 30, 15, 3, 2	lmhmlin 19035	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ a e. ( Base ` (Scalar ` S ) ) /\ ( `' F ` b ) e. X ) -> ( F ` ( a ( .s ` S ) ( `' F ` b ) ) ) = ( a ( .s ` T ) ( F ` ( `' F ` b ) ) ) )
32	19, 23, 29, 31	syl3anc 1326	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( F ` ( a ( .s ` S ) ( `' F ` b ) ) ) = ( a ( .s ` T ) ( F ` ( `' F ` b ) ) ) )
33		f1ocnvfv2 6533	. . . . . . 7 |- ( ( F : X -1-1-onto-> Y /\ b e. Y ) -> ( F ` ( `' F ` b ) ) = b )
34	33	ad2ant2l 782	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( F ` ( `' F ` b ) ) = b )
35	34	oveq2d 6666	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( a ( .s ` T ) ( F ` ( `' F ` b ) ) ) = ( a ( .s ` T ) b ) )
36	32, 35	eqtrd 2656	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( F ` ( a ( .s ` S ) ( `' F ` b ) ) ) = ( a ( .s ` T ) b ) )
37		simplr 792	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> F : X -1-1-onto-> Y )
38	10	adantr 481	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> S e. LMod )
39	15, 5, 3, 30	lmodvscl 18880	. . . . . 6 |- ( ( S e. LMod /\ a e. ( Base ` (Scalar ` S ) ) /\ ( `' F ` b ) e. X ) -> ( a ( .s ` S ) ( `' F ` b ) ) e. X )
40	38, 23, 29, 39	syl3anc 1326	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( a ( .s ` S ) ( `' F ` b ) ) e. X )
41		f1ocnvfv 6534	. . . . 5 |- ( ( F : X -1-1-onto-> Y /\ ( a ( .s ` S ) ( `' F ` b ) ) e. X ) -> ( ( F ` ( a ( .s ` S ) ( `' F ` b ) ) ) = ( a ( .s ` T ) b ) -> ( `' F ` ( a ( .s ` T ) b ) ) = ( a ( .s ` S ) ( `' F ` b ) ) ) )
42	37, 40, 41	syl2anc 693	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( ( F ` ( a ( .s ` S ) ( `' F ` b ) ) ) = ( a ( .s ` T ) b ) -> ( `' F ` ( a ( .s ` T ) b ) ) = ( a ( .s ` S ) ( `' F ` b ) ) ) )
43	36, 42	mpd 15	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. Y ) ) -> ( `' F ` ( a ( .s ` T ) b ) ) = ( a ( .s ` S ) ( `' F ` b ) ) )
44	1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43	islmhmd 19039	. 2 |- ( ( F e. ( S LMHom T ) /\ F : X -1-1-onto-> Y ) -> `' F e. ( T LMHom S ) )
45	15, 1	lmhmf 19034	. . . . 5 |- ( F e. ( S LMHom T ) -> F : X --> Y )
46		ffn 6045	. . . . 5 |- ( F : X --> Y -> F Fn X )
47	45, 46	syl 17	. . . 4 |- ( F e. ( S LMHom T ) -> F Fn X )
48	47	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> F Fn X )
49	1, 15	lmhmf 19034	. . . . 5 |- ( `' F e. ( T LMHom S ) -> `' F : Y --> X )
50	49	adantl 482	. . . 4 |- ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> `' F : Y --> X )
51		ffn 6045	. . . 4 |- ( `' F : Y --> X -> `' F Fn Y )
52	50, 51	syl 17	. . 3 |- ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> `' F Fn Y )
53		dff1o4 6145	. . 3 |- ( F : X -1-1-onto-> Y <-> ( F Fn X /\ `' F Fn Y ) )
54	48, 52, 53	sylanbrc 698	. 2 |- ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> F : X -1-1-onto-> Y )
55	44, 54	impbida 877	1 |- ( F e. ( S LMHom T ) -> ( F : X -1-1-onto-> Y <-> `' F e. ( T LMHom S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   `'ccnv 5113   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   GrpHom cghm 17657   LModclmod 18863   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  islmim2  19066