Theorem lmodpropd 14362

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 2231	. 2 |- (Scalar ` K ) = (Scalar ` K )
4		eqid 2231	. 2 |- (Scalar ` L ) = (Scalar ` L )
5		lmodpropd.6	. . 3 |- P = ( Base ` F )
6		lmodpropd.4	. . . 4 |- ( ph -> F = (Scalar ` K ) )
7	6	fveq2d 5643	. . 3 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` K ) ) )
8	5, 7	eqtrid 2276	. 2 |- ( ph -> P = ( Base ` (Scalar ` K ) ) )
9		lmodpropd.5	. . . 4 |- ( ph -> F = (Scalar ` L ) )
10	9	fveq2d 5643	. . 3 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` L ) ) )
11	5, 10	eqtrid 2276	. 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 2266	. . . . 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 5643	. . 3 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( +g ` (Scalar ` K ) ) = ( +g ` (Scalar ` L ) ) )
16	15	oveqd 6034	. 2 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( x ( +g ` (Scalar ` K ) ) y ) = ( x ( +g ` (Scalar ` L ) ) y ) )
17	14	fveq2d 5643	. . 3 |- ( ( ph /\ ( x e. P /\ y e. P ) ) -> ( .r ` (Scalar ` K ) ) = ( .r ` (Scalar ` L ) ) )
18	17	oveqd 6034	. 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 14361	1 |- ( ph -> ( K e. LMod <-> L e. LMod ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160  Scalarcsca 13162   .scvsca 13163   LModclmod 14300

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-mgp 13933  df-ur 13972  df-ring 14010  df-lmod 14302

This theorem is referenced by: (None)