Theorem lmodfopnelem1 18899

Description: Lemma 1 for lmodfopne 18901. (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

Step	Hyp	Ref	Expression
1		lmodfopne.v	. . . . 5 |- V = ( Base ` W )
2		lmodfopne.s	. . . . 5 |- S = (Scalar ` W )
3		lmodfopne.k	. . . . 5 |- K = ( Base ` S )
4		lmodfopne.t	. . . . 5 |- .x. = ( .sf ` W )
5	1, 2, 3, 4	lmodscaf 18885	. . . 4 |- ( W e. LMod -> .x. : ( K X. V ) --> V )
6	5	ffnd 6046	. . 3 |- ( W e. LMod -> .x. Fn ( K X. V ) )
7		lmodfopne.a	. . . . 5 |- .+ = ( +f ` W )
8	1, 7	plusffn 17250	. . . 4 |- .+ Fn ( V X. V )
9		fneq1 5979	. . . . . . . . . . 11 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) <-> .x. Fn ( V X. V ) ) )
10		fndmu 5992	. . . . . . . . . . . 12 |- ( ( .x. Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( V X. V ) = ( K X. V ) )
11	10	ex 450	. . . . . . . . . . 11 |- ( .x. Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( V X. V ) = ( K X. V ) ) )
12	9, 11	syl6bi 243	. . . . . . . . . 10 |- ( .+ = .x. -> ( .+ Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( V X. V ) = ( K X. V ) ) ) )
13	12	com13 88	. . . . . . . . 9 |- ( .x. Fn ( K X. V ) -> ( .+ Fn ( V X. V ) -> ( .+ = .x. -> ( V X. V ) = ( K X. V ) ) ) )
14	13	impcom 446	. . . . . . . 8 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( .+ = .x. -> ( V X. V ) = ( K X. V ) ) )
15	1	lmodbn0 18873	. . . . . . . . . . 11 |- ( W e. LMod -> V =/= (/) )
16		xp11 5569	. . . . . . . . . . 11 |- ( ( V =/= (/) /\ V =/= (/) ) -> ( ( V X. V ) = ( K X. V ) <-> ( V = K /\ V = V ) ) )
17	15, 15, 16	syl2anc 693	. . . . . . . . . 10 |- ( W e. LMod -> ( ( V X. V ) = ( K X. V ) <-> ( V = K /\ V = V ) ) )
18	17	simprbda 653	. . . . . . . . 9 |- ( ( W e. LMod /\ ( V X. V ) = ( K X. V ) ) -> V = K )
19	18	expcom 451	. . . . . . . 8 |- ( ( V X. V ) = ( K X. V ) -> ( W e. LMod -> V = K ) )
20	14, 19	syl6 35	. . . . . . 7 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( .+ = .x. -> ( W e. LMod -> V = K ) ) )
21	20	com23 86	. . . . . 6 |- ( ( .+ Fn ( V X. V ) /\ .x. Fn ( K X. V ) ) -> ( W e. LMod -> ( .+ = .x. -> V = K ) ) )
22	21	ex 450	. . . . 5 |- ( .+ Fn ( V X. V ) -> ( .x. Fn ( K X. V ) -> ( W e. LMod -> ( .+ = .x. -> V = K ) ) ) )
23	22	com23 86	. . . 4 |- ( .+ Fn ( V X. V ) -> ( W e. LMod -> ( .x. Fn ( K X. V ) -> ( .+ = .x. -> V = K ) ) ) )
24	8, 23	ax-mp 5	. . 3 |- ( W e. LMod -> ( .x. Fn ( K X. V ) -> ( .+ = .x. -> V = K ) ) )
25	6, 24	mpd 15	. 2 |- ( W e. LMod -> ( .+ = .x. -> V = K ) )
26	25	imp 445	1 |- ( ( W e. LMod /\ .+ = .x. ) -> V = K )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   X. cxp 5112   Fn wfn 5883   ` cfv 5888   Basecbs 15857  Scalarcsca 15944   +fcplusf 17239   LModclmod 18863   .sfcscaf 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-0g 16102  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-lmod 18865  df-scaf 18866

This theorem is referenced by:  lmodfopnelem2  18900