Theorem lmodfopnelem1 13508

Description: Lemma 1 for lmodfopne 13510. (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 12802	. . 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 13493	. . 3 |- ( W e. LMod -> .x. Fn ( K X. V ) )
8		fneq1 5316	. . . . . . . . . 10 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) <-> .x. Fn ( V X. V ) ) )
9		fndmu 5329	. . . . . . . . . . 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 13478	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Grp )
15		eqid 2187	. . . . . . . . . . . 12 |- ( 0g ` W ) = ( 0g ` W )
16	1, 15	grpidcl 12925	. . . . . . . . . . 11 |- ( W e. Grp -> ( 0g ` W ) e. V )
17		elex2 2765	. . . . . . . . . . 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 5079	. . . . . . . . . 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 1363   E.wex 1502   e. wcel 2158   X. cxp 4636   Fn wfn 5223   ` cfv 5228   Basecbs 12475  Scalarcsca 12553   0gc0g 12722   +fcplusf 12790   Grpcgrp 12898   LModclmod 13471   .sfcscaf 13472

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-mulr 12564  df-sca 12566  df-vsca 12567  df-0g 12724  df-plusf 12792  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-lmod 13473  df-scaf 13474

This theorem is referenced by:  lmodfopnelem2  13509