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Theorem lmodlema 18868

Description: Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
islmod.v
islmod.a
islmod.s
islmod.f	Scalar
islmod.k
islmod.p
islmod.t
islmod.u

**Assertion**
Ref	Expression
lmodlema	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Proof of Theorem lmodlema Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		islmod.v	. . . . . 6
2		islmod.a	. . . . . 6
3		islmod.s	. . . . . 6
4		islmod.f	. . . . . 6 Scalar
5		islmod.k	. . . . . 6
6		islmod.p	. . . . . 6
7		islmod.t	. . . . . 6
8		islmod.u	. . . . . 6
9	1, 2, 3, 4, 5, 6, 7, 8	islmod 18867	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10	9	simp3bi 1078	. . . 4 $/\$ $/\$ $/\$ $/\$
11		oveq1 6657	. . . . . . . . . 10
12	11	oveq1d 6665	. . . . . . . . 9
13		oveq1 6657	. . . . . . . . . 10
14	13	oveq1d 6665	. . . . . . . . 9
15	12, 14	eqeq12d 2637	. . . . . . . 8
16	15	3anbi3d 1405	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17		oveq1 6657	. . . . . . . . . 10
18	17	oveq1d 6665	. . . . . . . . 9
19		oveq1 6657	. . . . . . . . 9
20	18, 19	eqeq12d 2637	. . . . . . . 8
21	20	anbi1d 741	. . . . . . 7 $/\$ $/\$
22	16, 21	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	22	2ralbidv 2989	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		oveq1 6657	. . . . . . . . 9
25	24	eleq1d 2686	. . . . . . . 8
26		oveq1 6657	. . . . . . . . 9
27		oveq1 6657	. . . . . . . . . 10
28	24, 27	oveq12d 6668	. . . . . . . . 9
29	26, 28	eqeq12d 2637	. . . . . . . 8
30		oveq2 6658	. . . . . . . . . 10
31	30	oveq1d 6665	. . . . . . . . 9
32	24	oveq2d 6666	. . . . . . . . 9
33	31, 32	eqeq12d 2637	. . . . . . . 8
34	25, 29, 33	3anbi123d 1399	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
35		oveq2 6658	. . . . . . . . . 10
36	35	oveq1d 6665	. . . . . . . . 9
37	24	oveq2d 6666	. . . . . . . . 9
38	36, 37	eqeq12d 2637	. . . . . . . 8
39	38	anbi1d 741	. . . . . . 7 $/\$ $/\$
40	34, 39	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	2ralbidv 2989	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	23, 41	rspc2v 3322	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	10, 42	mpan9 486	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		oveq2 6658	. . . . . . . 8
45	44	oveq2d 6666	. . . . . . 7
46		oveq2 6658	. . . . . . . 8
47	46	oveq2d 6666	. . . . . . 7
48	45, 47	eqeq12d 2637	. . . . . 6
49	48	3anbi2d 1404	. . . . 5 $/\$ $/\$ $/\$ $/\$
50	49	anbi1d 741	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51		oveq2 6658	. . . . . . 7
52	51	eleq1d 2686	. . . . . 6
53		oveq1 6657	. . . . . . . 8
54	53	oveq2d 6666	. . . . . . 7
55	51	oveq1d 6665	. . . . . . 7
56	54, 55	eqeq12d 2637	. . . . . 6
57		oveq2 6658	. . . . . . 7
58		oveq2 6658	. . . . . . . 8
59	58, 51	oveq12d 6668	. . . . . . 7
60	57, 59	eqeq12d 2637	. . . . . 6
61	52, 56, 60	3anbi123d 1399	. . . . 5 $/\$ $/\$ $/\$ $/\$
62		oveq2 6658	. . . . . . 7
63	51	oveq2d 6666	. . . . . . 7
64	62, 63	eqeq12d 2637	. . . . . 6
65		oveq2 6658	. . . . . . 7
66		id 22	. . . . . . 7
67	65, 66	eqeq12d 2637	. . . . . 6
68	64, 67	anbi12d 747	. . . . 5 $/\$ $/\$
69	61, 68	anbi12d 747	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	50, 69	rspc2v 3322	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	43, 70	syl5com 31	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	3impia 1261	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

cfv 5888 (class class class)co 6650

cbs 15857

cplusg 15941

cmulr 15942 Scalarcsca 15944

cvsca 15945

cgrp 17422

cur 18501

crg 18547

clmod 18863

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-lmod 18865

This theorem is referenced by: lmodvscl 18880 lmodvsdi 18886 lmodvsdir 18887 lmodvsass 18888 lmodvs1 18891

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