Theorem lmhmpropd 19073

Description: Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
lmhmpropd.a	|- ( ph -> B = ( Base ` J ) )
lmhmpropd.b	|- ( ph -> C = ( Base ` K ) )
lmhmpropd.c	|- ( ph -> B = ( Base ` L ) )
lmhmpropd.d	|- ( ph -> C = ( Base ` M ) )
lmhmpropd.1	|- ( ph -> F = (Scalar ` J ) )
lmhmpropd.2	|- ( ph -> G = (Scalar ` K ) )
lmhmpropd.3	|- ( ph -> F = (Scalar ` L ) )
lmhmpropd.4	|- ( ph -> G = (Scalar ` M ) )
lmhmpropd.p	|- P = ( Base ` F )
lmhmpropd.q	|- Q = ( Base ` G )
lmhmpropd.e	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
lmhmpropd.f	|- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
lmhmpropd.g	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` J ) y ) = ( x ( .s ` L ) y ) )
lmhmpropd.h	|- ( ( ph /\ ( x e. Q /\ y e. C ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` M ) y ) )

Assertion

Ref	Expression
lmhmpropd	|- ( ph -> ( J LMHom K ) = ( L LMHom M ) )

Distinct variable groups:   x, y, C   x, J, y   x, K, y   x, L, y   x, M, y   x, P, y   ph, x, y   x, B, y   x, Q, y

Allowed substitution hints:   F( x, y)   G( x, y)

Proof of Theorem lmhmpropd

Dummy variables z w f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmpropd.a	. . . . . 6 |- ( ph -> B = ( Base ` J ) )
2		lmhmpropd.c	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
3		lmhmpropd.e	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
4		lmhmpropd.1	. . . . . 6 |- ( ph -> F = (Scalar ` J ) )
5		lmhmpropd.3	. . . . . 6 |- ( ph -> F = (Scalar ` L ) )
6		lmhmpropd.p	. . . . . 6 |- P = ( Base ` F )
7		lmhmpropd.g	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` J ) y ) = ( x ( .s ` L ) y ) )
8	1, 2, 3, 4, 5, 6, 7	lmodpropd 18926	. . . . 5 |- ( ph -> ( J e. LMod <-> L e. LMod ) )
9		lmhmpropd.b	. . . . . 6 |- ( ph -> C = ( Base ` K ) )
10		lmhmpropd.d	. . . . . 6 |- ( ph -> C = ( Base ` M ) )
11		lmhmpropd.f	. . . . . 6 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
12		lmhmpropd.2	. . . . . 6 |- ( ph -> G = (Scalar ` K ) )
13		lmhmpropd.4	. . . . . 6 |- ( ph -> G = (Scalar ` M ) )
14		lmhmpropd.q	. . . . . 6 |- Q = ( Base ` G )
15		lmhmpropd.h	. . . . . 6 |- ( ( ph /\ ( x e. Q /\ y e. C ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` M ) y ) )
16	9, 10, 11, 12, 13, 14, 15	lmodpropd 18926	. . . . 5 |- ( ph -> ( K e. LMod <-> M e. LMod ) )
17	8, 16	anbi12d 747	. . . 4 |- ( ph -> ( ( J e. LMod /\ K e. LMod ) <-> ( L e. LMod /\ M e. LMod ) ) )
18	7	oveqrspc2v 6673	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. P /\ w e. B ) ) -> ( z ( .s ` J ) w ) = ( z ( .s ` L ) w ) )
19	18	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( z ( .s ` J ) w ) = ( z ( .s ` L ) w ) )
20	19	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( f ` ( z ( .s ` J ) w ) ) = ( f ` ( z ( .s ` L ) w ) ) )
21		simpll 790	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ph )
22		simprl 794	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> z e. P )
23		simplrr 801	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> G = F )
24	23	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( Base ` G ) = ( Base ` F ) )
25	24, 14, 6	3eqtr4g 2681	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> Q = P )
26	22, 25	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> z e. Q )
27		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> f e. ( J GrpHom K ) )
28		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` J ) = ( Base ` J )
29		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` K ) = ( Base ` K )
30	28, 29	ghmf 17664	. . . . . . . . . . . . 13 |- ( f e. ( J GrpHom K ) -> f : ( Base ` J ) --> ( Base ` K ) )
31	27, 30	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> f : ( Base ` J ) --> ( Base ` K ) )
32		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> w e. B )
33	21, 1	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> B = ( Base ` J ) )
34	32, 33	eleqtrd 2703	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> w e. ( Base ` J ) )
35	31, 34	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( f ` w ) e. ( Base ` K ) )
36	21, 9	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> C = ( Base ` K ) )
37	35, 36	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( f ` w ) e. C )
38	15	oveqrspc2v 6673	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q /\ ( f ` w ) e. C ) ) -> ( z ( .s ` K ) ( f ` w ) ) = ( z ( .s ` M ) ( f ` w ) ) )
39	21, 26, 37, 38	syl12anc 1324	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( z ( .s ` K ) ( f ` w ) ) = ( z ( .s ` M ) ( f ` w ) ) )
40	20, 39	eqeq12d 2637	. . . . . . . 8 |- ( ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) /\ ( z e. P /\ w e. B ) ) -> ( ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) <-> ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) )
41	40	2ralbidva 2988	. . . . . . 7 |- ( ( ph /\ ( f e. ( J GrpHom K ) /\ G = F ) ) -> ( A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) <-> A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) )
42	41	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( ( f e. ( J GrpHom K ) /\ G = F ) /\ A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) <-> ( ( f e. ( J GrpHom K ) /\ G = F ) /\ A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) )
43		df-3an 1039	. . . . . 6 |- ( ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) <-> ( ( f e. ( J GrpHom K ) /\ G = F ) /\ A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) )
44		df-3an 1039	. . . . . 6 |- ( ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) <-> ( ( f e. ( J GrpHom K ) /\ G = F ) /\ A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) )
45	42, 43, 44	3bitr4g 303	. . . . 5 |- ( ph -> ( ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) <-> ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) )
46	12, 4	eqeq12d 2637	. . . . . 6 |- ( ph -> ( G = F <-> (Scalar ` K ) = (Scalar ` J ) ) )
47	4	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` J ) ) )
48	6, 47	syl5eq 2668	. . . . . . 7 |- ( ph -> P = ( Base ` (Scalar ` J ) ) )
49	1	raleqdv 3144	. . . . . . 7 |- ( ph -> ( A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) <-> A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) )
50	48, 49	raleqbidv 3152	. . . . . 6 |- ( ph -> ( A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) <-> A. z e. ( Base ` (Scalar ` J ) ) A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) )
51	46, 50	3anbi23d 1402	. . . . 5 |- ( ph -> ( ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) <-> ( f e. ( J GrpHom K ) /\ (Scalar ` K ) = (Scalar ` J ) /\ A. z e. ( Base ` (Scalar ` J ) ) A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) ) )
52	1, 9, 2, 10, 3, 11	ghmpropd 17698	. . . . . . 7 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
53	52	eleq2d 2687	. . . . . 6 |- ( ph -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
54	13, 5	eqeq12d 2637	. . . . . 6 |- ( ph -> ( G = F <-> (Scalar ` M ) = (Scalar ` L ) ) )
55	5	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` L ) ) )
56	6, 55	syl5eq 2668	. . . . . . 7 |- ( ph -> P = ( Base ` (Scalar ` L ) ) )
57	2	raleqdv 3144	. . . . . . 7 |- ( ph -> ( A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) <-> A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) )
58	56, 57	raleqbidv 3152	. . . . . 6 |- ( ph -> ( A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) <-> A. z e. ( Base ` (Scalar ` L ) ) A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) )
59	53, 54, 58	3anbi123d 1399	. . . . 5 |- ( ph -> ( ( f e. ( J GrpHom K ) /\ G = F /\ A. z e. P A. w e. B ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) <-> ( f e. ( L GrpHom M ) /\ (Scalar ` M ) = (Scalar ` L ) /\ A. z e. ( Base ` (Scalar ` L ) ) A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) )
60	45, 51, 59	3bitr3d 298	. . . 4 |- ( ph -> ( ( f e. ( J GrpHom K ) /\ (Scalar ` K ) = (Scalar ` J ) /\ A. z e. ( Base ` (Scalar ` J ) ) A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) <-> ( f e. ( L GrpHom M ) /\ (Scalar ` M ) = (Scalar ` L ) /\ A. z e. ( Base ` (Scalar ` L ) ) A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) )
61	17, 60	anbi12d 747	. . 3 |- ( ph -> ( ( ( J e. LMod /\ K e. LMod ) /\ ( f e. ( J GrpHom K ) /\ (Scalar ` K ) = (Scalar ` J ) /\ A. z e. ( Base ` (Scalar ` J ) ) A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) ) <-> ( ( L e. LMod /\ M e. LMod ) /\ ( f e. ( L GrpHom M ) /\ (Scalar ` M ) = (Scalar ` L ) /\ A. z e. ( Base ` (Scalar ` L ) ) A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) ) )
62		eqid 2622	. . . 4 |- (Scalar ` J ) = (Scalar ` J )
63		eqid 2622	. . . 4 |- (Scalar ` K ) = (Scalar ` K )
64		eqid 2622	. . . 4 |- ( Base ` (Scalar ` J ) ) = ( Base ` (Scalar ` J ) )
65		eqid 2622	. . . 4 |- ( .s ` J ) = ( .s ` J )
66		eqid 2622	. . . 4 |- ( .s ` K ) = ( .s ` K )
67	62, 63, 64, 28, 65, 66	islmhm 19027	. . 3 |- ( f e. ( J LMHom K ) <-> ( ( J e. LMod /\ K e. LMod ) /\ ( f e. ( J GrpHom K ) /\ (Scalar ` K ) = (Scalar ` J ) /\ A. z e. ( Base ` (Scalar ` J ) ) A. w e. ( Base ` J ) ( f ` ( z ( .s ` J ) w ) ) = ( z ( .s ` K ) ( f ` w ) ) ) ) )
68		eqid 2622	. . . 4 |- (Scalar ` L ) = (Scalar ` L )
69		eqid 2622	. . . 4 |- (Scalar ` M ) = (Scalar ` M )
70		eqid 2622	. . . 4 |- ( Base ` (Scalar ` L ) ) = ( Base ` (Scalar ` L ) )
71		eqid 2622	. . . 4 |- ( Base ` L ) = ( Base ` L )
72		eqid 2622	. . . 4 |- ( .s ` L ) = ( .s ` L )
73		eqid 2622	. . . 4 |- ( .s ` M ) = ( .s ` M )
74	68, 69, 70, 71, 72, 73	islmhm 19027	. . 3 |- ( f e. ( L LMHom M ) <-> ( ( L e. LMod /\ M e. LMod ) /\ ( f e. ( L GrpHom M ) /\ (Scalar ` M ) = (Scalar ` L ) /\ A. z e. ( Base ` (Scalar ` L ) ) A. w e. ( Base ` L ) ( f ` ( z ( .s ` L ) w ) ) = ( z ( .s ` M ) ( f ` w ) ) ) ) )
75	61, 67, 74	3bitr4g 303	. 2 |- ( ph -> ( f e. ( J LMHom K ) <-> f e. ( L LMHom M ) ) )
76	75	eqrdv 2620	1 |- ( ph -> ( J LMHom K ) = ( L LMHom M ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  phlpropd  20000