Theorem lmodfopnelem1 14086

Description: Lemma 1 for lmodfopne 14088. (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 13197	. . 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 14071	. . 3 |- ( W e. LMod -> .x. Fn ( K X. V ) )
8		fneq1 5362	. . . . . . . . . 10 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) <-> .x. Fn ( V X. V ) ) )
9		fndmu 5377	. . . . . . . . . . 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 14056	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Grp )
15		eqid 2205	. . . . . . . . . . . 12 |- ( 0g ` W ) = ( 0g ` W )
16	1, 15	grpidcl 13361	. . . . . . . . . . 11 |- ( W e. Grp -> ( 0g ` W ) e. V )
17		elex2 2788	. . . . . . . . . . 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 5121	. . . . . . . . . 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 1373   E.wex 1515   e. wcel 2176   X. cxp 4673   Fn wfn 5266   ` cfv 5271   Basecbs 12832  Scalarcsca 12912   0gc0g 13088   +fcplusf 13185   Grpcgrp 13332   LModclmod 14049   .sfcscaf 14050

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-0g 13090  df-plusf 13187  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-lmod 14051  df-scaf 14052

This theorem is referenced by:  lmodfopnelem2  14087