Theorem lmodfopne 14405

Description: The (functionalized) operations of a left module (over a nonzero ring) cannot be identical. (Contributed by NM, 31-May-2008.) (Revised 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 )
lmodfopne.0	|- .0. = ( 0g ` S )
lmodfopne.1	|- .1. = ( 1r ` S )

Assertion

Ref	Expression
lmodfopne	|- ( ( W e. LMod /\ .1. =/= .0. ) -> .+ =/= .x. )

Proof of Theorem lmodfopne

Step	Hyp	Ref	Expression
1		lmodfopne.t	. . . . . 6 |- .x. = ( .sf ` W )
2		lmodfopne.a	. . . . . 6 |- .+ = ( +f ` W )
3		lmodfopne.v	. . . . . 6 |- V = ( Base ` W )
4		lmodfopne.s	. . . . . 6 |- S = (Scalar ` W )
5		lmodfopne.k	. . . . . 6 |- K = ( Base ` S )
6		lmodfopne.0	. . . . . 6 |- .0. = ( 0g ` S )
7		lmodfopne.1	. . . . . 6 |- .1. = ( 1r ` S )
8	1, 2, 3, 4, 5, 6, 7	lmodfopnelem2 14404	. . . . 5 |- ( ( W e. LMod /\ .+ = .x. ) -> ( .0. e. V /\ .1. e. V ) )
9		simpll 527	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> W e. LMod )
10		simpl 109	. . . . . . . . 9 |- ( ( .0. e. V /\ .1. e. V ) -> .0. e. V )
11	10	adantl 277	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .0. e. V )
12		eqid 2231	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
13	3, 12	lmod0vcl 14396	. . . . . . . . 9 |- ( W e. LMod -> ( 0g ` W ) e. V )
14	13	ad2antrr 488	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( 0g ` W ) e. V )
15		eqid 2231	. . . . . . . . . 10 |- ( +g ` W ) = ( +g ` W )
16	3, 15, 2	plusfvalg 13509	. . . . . . . . 9 |- ( ( W e. LMod /\ .0. e. V /\ ( 0g ` W ) e. V ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. ( +g ` W ) ( 0g ` W ) ) )
17	16	eqcomd 2237	. . . . . . . 8 |- ( ( W e. LMod /\ .0. e. V /\ ( 0g ` W ) e. V ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .+ ( 0g ` W ) ) )
18	9, 11, 14, 17	syl3anc 1274	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .+ ( 0g ` W ) ) )
19		oveq 6034	. . . . . . . 8 |- ( .+ = .x. -> ( .0. .+ ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
20	19	ad2antlr 489	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
21	18, 20	eqtrd 2264	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
22		lmodgrp 14373	. . . . . . . 8 |- ( W e. LMod -> W e. Grp )
23	22	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ .+ = .x. ) -> W e. Grp )
24	3, 15, 12	grprid 13678	. . . . . . 7 |- ( ( W e. Grp /\ .0. e. V ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = .0. )
25	23, 10, 24	syl2an 289	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = .0. )
26	4, 5, 6	lmod0cl 14393	. . . . . . . . 9 |- ( W e. LMod -> .0. e. K )
27	26	ad2antrr 488	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .0. e. K )
28		eqid 2231	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
29	3, 4, 5, 1, 28	scafvalg 14386	. . . . . . . 8 |- ( ( W e. LMod /\ .0. e. K /\ ( 0g ` W ) e. V ) -> ( .0. .x. ( 0g ` W ) ) = ( .0. ( .s ` W ) ( 0g ` W ) ) )
30	9, 27, 14, 29	syl3anc 1274	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .x. ( 0g ` W ) ) = ( .0. ( .s ` W ) ( 0g ` W ) ) )
31	26	ancli 323	. . . . . . . . 9 |- ( W e. LMod -> ( W e. LMod /\ .0. e. K ) )
32	31	ad2antrr 488	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( W e. LMod /\ .0. e. K ) )
33	4, 28, 5, 12	lmodvs0 14401	. . . . . . . 8 |- ( ( W e. LMod /\ .0. e. K ) -> ( .0. ( .s ` W ) ( 0g ` W ) ) = ( 0g ` W ) )
34	32, 33	syl 14	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( .s ` W ) ( 0g ` W ) ) = ( 0g ` W ) )
35		simpr 110	. . . . . . . . . 10 |- ( ( .0. e. V /\ .1. e. V ) -> .1. e. V )
36	3, 15, 12	grprid 13678	. . . . . . . . . 10 |- ( ( W e. Grp /\ .1. e. V ) -> ( .1. ( +g ` W ) ( 0g ` W ) ) = .1. )
37	23, 35, 36	syl2an 289	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. ( +g ` W ) ( 0g ` W ) ) = .1. )
38	4, 5, 7	lmod1cl 14394	. . . . . . . . . . . 12 |- ( W e. LMod -> .1. e. K )
39	38	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .1. e. K )
40	35	adantl 277	. . . . . . . . . . 11 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .1. e. V )
41	3, 4, 5, 1, 28	scafvalg 14386	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. K /\ .1. e. V ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
42	9, 39, 40, 41	syl3anc 1274	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
43	3, 4, 28, 7	lmodvs1 14395	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V ) -> ( .1. ( .s ` W ) .1. ) = .1. )
44	43	ad2ant2rl 511	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. ( .s ` W ) .1. ) = .1. )
45	42, 44	eqtrd 2264	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = .1. )
46		oveq 6034	. . . . . . . . . . . 12 |- ( .+ = .x. -> ( .1. .+ .1. ) = ( .1. .x. .1. ) )
47	46	eqcomd 2237	. . . . . . . . . . 11 |- ( .+ = .x. -> ( .1. .x. .1. ) = ( .1. .+ .1. ) )
48	47	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. .+ .1. ) )
49	3, 15, 2	plusfvalg 13509	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V /\ .1. e. V ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
50	9, 40, 40, 49	syl3anc 1274	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
51	48, 50	eqtrd 2264	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( +g ` W ) .1. ) )
52	37, 45, 51	3eqtr2d 2270	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. ( +g ` W ) ( 0g ` W ) ) = ( .1. ( +g ` W ) .1. ) )
53	22	ad2antrr 488	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> W e. Grp )
54	3, 15	grplcan 13708	. . . . . . . . 9 |- ( ( W e. Grp /\ ( ( 0g ` W ) e. V /\ .1. e. V /\ .1. e. V ) ) -> ( ( .1. ( +g ` W ) ( 0g ` W ) ) = ( .1. ( +g ` W ) .1. ) <-> ( 0g ` W ) = .1. ) )
55	53, 14, 40, 40, 54	syl13anc 1276	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( ( .1. ( +g ` W ) ( 0g ` W ) ) = ( .1. ( +g ` W ) .1. ) <-> ( 0g ` W ) = .1. ) )
56	52, 55	mpbid 147	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( 0g ` W ) = .1. )
57	30, 34, 56	3eqtrd 2268	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .x. ( 0g ` W ) ) = .1. )
58	21, 25, 57	3eqtr3rd 2273	. . . . 5 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .1. = .0. )
59	8, 58	mpdan 421	. . . 4 |- ( ( W e. LMod /\ .+ = .x. ) -> .1. = .0. )
60	59	ex 115	. . 3 |- ( W e. LMod -> ( .+ = .x. -> .1. = .0. ) )
61	60	necon3d 2447	. 2 |- ( W e. LMod -> ( .1. =/= .0. -> .+ =/= .x. ) )
62	61	imp 124	1 |- ( ( W e. LMod /\ .1. =/= .0. ) -> .+ =/= .x. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223  Scalarcsca 13226   .scvsca 13227   0gc0g 13402   +fcplusf 13499   Grpcgrp 13646   1rcur 14036   LModclmod 14366   .sfcscaf 14367

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-0g 13404  df-plusf 13501  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-mgp 13998  df-ur 14037  df-ring 14075  df-lmod 14368  df-scaf 14369

This theorem is referenced by: (None)