Theorem lmodfopnelem1 14636

Description: Lemma 1 for lmodfopne 14638. (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 13665	. . 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 14621	. . 3 |- ( W e. LMod -> .x. Fn ( K X. V ) )
8		fneq1 5467	. . . . . . . . . 10 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) <-> .x. Fn ( V X. V ) ) )
9		fndmu 5482	. . . . . . . . . . 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 14606	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Grp )
15		eqid 2238	. . . . . . . . . . . 12 |- ( 0g ` W ) = ( 0g ` W )
16	1, 15	grpidcl 13814	. . . . . . . . . . 11 |- ( W e. Grp -> ( 0g ` W ) e. V )
17		elex2 2838	. . . . . . . . . . 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 5224	. . . . . . . . . 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 415	. . . . . . . . 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 1402   E.wex 1545   e. wcel 2209   X. cxp 4770   Fn wfn 5370   ` cfv 5375   Basecbs 13333  Scalarcsca 13414   0gc0g 13590   +fcplusf 13653   Grpcgrp 13785   LModclmod 14599   .sfcscaf 14600

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8263  ax-resscn 8264  ax-1re 8266  ax-addrcl 8269

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-5 9348  df-6 9349  df-ndx 13336  df-slot 13337  df-base 13339  df-plusg 13424  df-mulr 13425  df-sca 13427  df-vsca 13428  df-0g 13592  df-plusf 13655  df-mgm 13656  df-sgrp 13697  df-mnd 13710  df-grp 13788  df-lmod 14601  df-scaf 14602

This theorem is referenced by:  lmodfopnelem2  14637