Theorem lmodfopne 14663

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 14662	. . . . 5 |- ( ( W e. LMod /\ .+ = .x. ) -> ( .0. e. V /\ .1. e. V ) )
9		simpll 531	. . . . . . . 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 2238	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
13	3, 12	lmod0vcl 14654	. . . . . . . . 9 |- ( W e. LMod -> ( 0g ` W ) e. V )
14	13	ad2antrr 492	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( 0g ` W ) e. V )
15		eqid 2238	. . . . . . . . . 10 |- ( +g ` W ) = ( +g ` W )
16	3, 15, 2	plusfvalg 13683	. . . . . . . . 9 |- ( ( W e. LMod /\ .0. e. V /\ ( 0g ` W ) e. V ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. ( +g ` W ) ( 0g ` W ) ) )
17	16	eqcomd 2244	. . . . . . . 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 1278	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .+ ( 0g ` W ) ) )
19		oveq 6091	. . . . . . . 8 |- ( .+ = .x. -> ( .0. .+ ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
20	19	ad2antlr 493	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
21	18, 20	eqtrd 2271	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
22		lmodgrp 14630	. . . . . . . 8 |- ( W e. LMod -> W e. Grp )
23	22	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ .+ = .x. ) -> W e. Grp )
24	3, 15, 12	grprid 13837	. . . . . . 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 14651	. . . . . . . . 9 |- ( W e. LMod -> .0. e. K )
27	26	ad2antrr 492	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .0. e. K )
28		eqid 2238	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
29	3, 4, 5, 1, 28	scafvalg 14644	. . . . . . . 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 1278	. . . . . . 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 492	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( W e. LMod /\ .0. e. K ) )
33	4, 28, 5, 12	lmodvs0 14659	. . . . . . . 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 13837	. . . . . . . . . 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 14652	. . . . . . . . . . . 12 |- ( W e. LMod -> .1. e. K )
39	38	ad2antrr 492	. . . . . . . . . . 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 14644	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. K /\ .1. e. V ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
42	9, 39, 40, 41	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
43	3, 4, 28, 7	lmodvs1 14653	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V ) -> ( .1. ( .s ` W ) .1. ) = .1. )
44	43	ad2ant2rl 515	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. ( .s ` W ) .1. ) = .1. )
45	42, 44	eqtrd 2271	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = .1. )
46		oveq 6091	. . . . . . . . . . . 12 |- ( .+ = .x. -> ( .1. .+ .1. ) = ( .1. .x. .1. ) )
47	46	eqcomd 2244	. . . . . . . . . . 11 |- ( .+ = .x. -> ( .1. .x. .1. ) = ( .1. .+ .1. ) )
48	47	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. .+ .1. ) )
49	3, 15, 2	plusfvalg 13683	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V /\ .1. e. V ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
50	9, 40, 40, 49	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
51	48, 50	eqtrd 2271	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( +g ` W ) .1. ) )
52	37, 45, 51	3eqtr2d 2277	. . . . . . . 8 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. ( +g ` W ) ( 0g ` W ) ) = ( .1. ( +g ` W ) .1. ) )
53	22	ad2antrr 492	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> W e. Grp )
54	3, 15	grplcan 13867	. . . . . . . . 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 1280	. . . . . . . 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 2275	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .x. ( 0g ` W ) ) = .1. )
58	21, 25, 57	3eqtr3rd 2280	. . . . 5 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> .1. = .0. )
59	8, 58	mpdan 425	. . . 4 |- ( ( W e. LMod /\ .+ = .x. ) -> .1. = .0. )
60	59	ex 115	. . 3 |- ( W e. LMod -> ( .+ = .x. -> .1. = .0. ) )
61	60	necon3d 2464	. 2 |- ( W e. LMod -> ( .1. =/= .0. -> .+ =/= .x. ) )
62	61	imp 124	1 |- ( ( W e. LMod /\ .1. =/= .0. ) -> .+ =/= .x. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431  Scalarcsca 13434   .scvsca 13435   0gc0g 13610   +fcplusf 13673   Grpcgrp 13805   1rcur 14262   LModclmod 14623   .sfcscaf 14624

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-0g 13612  df-plusf 13675  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-mgp 14218  df-ur 14263  df-ring 14302  df-lmod 14625  df-scaf 14626

This theorem is used by: (None)