Theorem lmodpropd 14353

Description: If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)

Hypotheses

Ref	Expression
lmodpropd.1	|- ( ph -> B = ( Base ` K ) )
lmodpropd.2	|- ( ph -> B = ( Base ` L ) )
lmodpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
lmodpropd.4	|- ( ph -> F = (Scalar ` K ) )
lmodpropd.5	|- ( ph -> F = (Scalar ` L ) )
lmodpropd.6	|- P = ( Base ` F )
lmodpropd.7	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )

Assertion

Ref	Expression
lmodpropd	|- ( ph -> ( K e. LMod <-> L e. LMod ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   x, P, y   ph, x, y

Allowed substitution hints:   F( x, y)

Proof of Theorem lmodpropd

Step	Hyp	Ref	Expression
1		lmodpropd.1	. 2 |- ( ph -> B = ( Base ` K ) )
2		lmodpropd.2	. 2 |- ( ph -> B = ( Base ` L ) )
3		eqid 2229	. 2 |- (Scalar ` K ) = (Scalar ` K )
4		eqid 2229	. 2 |- (Scalar ` L ) = (Scalar ` L )
5		lmodpropd.6	. . 3 |- P = ( Base ` F )
6		lmodpropd.4	. . . 4 |- ( ph -> F = (Scalar ` K ) )
7	6	fveq2d 5639	. . 3 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` K ) ) )
8	5, 7	eqtrid 2274	. 2 |- ( ph -> P = ( Base ` (Scalar ` K ) ) )
9		lmodpropd.5	. . . 4 |- ( ph -> F = (Scalar ` L ) )
10	9	fveq2d 5639	. . 3 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` L ) ) )
11	5, 10	eqtrid 2274	. 2 |- ( ph -> P = ( Base ` (Scalar ` L ) ) )
12		lmodpropd.3	. 2 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
13	6, 9	eqtr3d 2264	. . . . 5 |- ( ph -> (Scalar ` K ) = (Scalar ` L ) )
14	13	adantr 276	. . . 4 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> (Scalar ` K ) = (Scalar ` L ) )
15	14	fveq2d 5639	. . 3 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( +g ` (Scalar ` K ) ) = ( +g ` (Scalar ` L ) ) )
16	15	oveqd 6030	. 2 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( +g ` (Scalar ` K ) ) y ) = ( x ( +g ` (Scalar ` L ) ) y ) )
17	14	fveq2d 5639	. . 3 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( .r ` (Scalar ` K ) ) = ( .r ` (Scalar ` L ) ) )
18	17	oveqd 6030	. 2 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( .r ` (Scalar ` K ) ) y ) = ( x ( .r ` (Scalar ` L ) ) y ) )
19		lmodpropd.7	. 2 |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
20	1, 2, 3, 4, 8, 11, 12, 16, 18, 19	lmodprop2d 14352	1 |- ( ph -> ( K e. LMod <-> L e. LMod ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150   .rcmulr 13151  Scalarcsca 13153   .scvsca 13154   LModclmod 14291

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-sca 13166  df-vsca 13167  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-mgp 13924  df-ur 13963  df-ring 14001  df-lmod 14293

This theorem is referenced by: (None)