Theorem lmodfopnelem1 13880

Description: Lemma 1 for lmodfopne 13882. (Contributed by AV, 2-Oct-2021.)

Hypotheses

Ref	Expression
lmodfopne.t	|- .x. = ( .sf ` W )
lmodfopne.a	|- .+ = ( +f ` W )
lmodfopne.v	|- V = ( Base ` W )
lmodfopne.s	|- S = (Scalar ` W )
lmodfopne.k	|- K = ( Base ` S )

Assertion

Ref	Expression
lmodfopnelem1	|- ( ( W e. LMod /\ .+ = .x. ) -> V = K )

Proof of Theorem lmodfopnelem1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmodfopne.v	. . . 4 |- V = ( Base ` W )
2		lmodfopne.a	. . . 4 |- .+ = ( +f ` W )
3	1, 2	plusffng 13008	. . 3 |- ( W e. LMod -> .+ Fn ( V X. V ) )
4		lmodfopne.s	. . . 4 |- S = (Scalar ` W )
5		lmodfopne.k	. . . 4 |- K = ( Base ` S )
6		lmodfopne.t	. . . 4 |- .x. = ( .sf ` W )
7	1, 4, 5, 6	scaffng 13865	. . 3 |- ( W e. LMod -> .x. Fn ( K X. V ) )
8		fneq1 5346	. . . . . . . . . 10 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) <-> .x. Fn ( V X. V ) ) )
9		fndmu 5359	. . . . . . . . . . 11 |- ( ( .x. Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( V X. V ) = ( K X. V ) )
10	9	ex 115	. . . . . . . . . 10 |- ( .x. Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( V X. V ) = ( K X. V ) ) )
11	8, 10	biimtrdi 163	. . . . . . . . 9 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( V X. V ) = ( K X. V ) ) ) )
12	11	com13 80	. . . . . . . 8 |- ( .x. Fn ( K X. V ) -> ( .+ Fn ( V X. V ) -> ( .+ = .x. -> ( V X. V ) = ( K X. V ) ) ) )
13	12	impcom 125	. . . . . . 7 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( .+ = .x. -> ( V X. V ) = ( K X. V ) ) )
14		lmodgrp 13850	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Grp )
15		eqid 2196	. . . . . . . . . . . 12 |- ( 0g ` W ) = ( 0g ` W )
16	1, 15	grpidcl 13161	. . . . . . . . . . 11 |- ( W e. Grp -> ( 0g ` W ) e. V )
17		elex2 2779	. . . . . . . . . . 11 |- ( ( 0g ` W ) e. V -> E. w w e. V )
18	14, 16, 17	3syl 17	. . . . . . . . . 10 |- ( W e. LMod -> E. w w e. V )
19		xp11m 5108	. . . . . . . . . 10 |- ( ( E. w w e. V /\ E. w w e. V ) -> ( ( V X. V ) = ( K X. V ) <-> ( V = K /\ V = V ) ) )
20	18, 18, 19	syl2anc 411	. . . . . . . . 9 |- ( W e. LMod -> ( ( V X. V ) = ( K X. V ) <-> ( V = K /\ V = V ) ) )
21	20	simprbda 383	. . . . . . . 8 |- ( ( W e. LMod /\ ( V X. V ) = ( K X. V ) ) -> V = K )
22	21	expcom 116	. . . . . . 7 |- ( ( V X. V ) = ( K X. V ) -> ( W e. LMod -> V = K ) )
23	13, 22	syl6 33	. . . . . 6 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( .+ = .x. -> ( W e. LMod -> V = K ) ) )
24	23	com23 78	. . . . 5 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( W e. LMod -> ( .+ = .x. -> V = K ) ) )
25	24	ex 115	. . . 4 |- ( .+ Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( W e. LMod -> ( .+ = .x. -> V = K ) ) ) )
26	25	com3r 79	. . 3 |- ( W e. LMod -> ( .+ Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( .+ = .x. -> V = K ) ) ) )
27	3, 7, 26	mp2d 47	. 2 |- ( W e. LMod -> ( .+ = .x. -> V = K ) )
28	27	imp 124	1 |- ( ( W e. LMod /\ .+ = .x. ) -> V = K )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   X. cxp 4661   Fn wfn 5253   ` cfv 5258   Basecbs 12678  Scalarcsca 12758   0gc0g 12927   +fcplusf 12996   Grpcgrp 13132   LModclmod 13843   .sfcscaf 13844

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-0g 12929  df-plusf 12998  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-lmod 13845  df-scaf 13846

This theorem is referenced by:  lmodfopnelem2  13881