Theorem lmodfopne 13603

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 13602	. . . . 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 2189	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
13	3, 12	lmod0vcl 13594	. . . . . . . . 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 2189	. . . . . . . . . 10 |- ( +g ` W ) = ( +g ` W )
16	3, 15, 2	plusfvalg 12805	. . . . . . . . 9 |- ( ( W e. LMod /\ .0. e. V /\ ( 0g ` W ) e. V ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. ( +g ` W ) ( 0g ` W ) ) )
17	16	eqcomd 2195	. . . . . . . 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 1249	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .+ ( 0g ` W ) ) )
19		oveq 5897	. . . . . . . 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 2222	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
22		lmodgrp 13571	. . . . . . . 8 |- ( W e. LMod -> W e. Grp )
23	22	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ .+ = .x. ) -> W e. Grp )
24	3, 15, 12	grprid 12942	. . . . . . 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 13591	. . . . . . . . 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 2189	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
29	3, 4, 5, 1, 28	scafvalg 13584	. . . . . . . 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 1249	. . . . . . 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 13599	. . . . . . . 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 12942	. . . . . . . . . 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 13592	. . . . . . . . . . . 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 13584	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. K /\ .1. e. V ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
42	9, 39, 40, 41	syl3anc 1249	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
43	3, 4, 28, 7	lmodvs1 13593	. . . . . . . . . . 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 2222	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = .1. )
46		oveq 5897	. . . . . . . . . . . 12 |- ( .+ = .x. -> ( .1. .+ .1. ) = ( .1. .x. .1. ) )
47	46	eqcomd 2195	. . . . . . . . . . 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 12805	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V /\ .1. e. V ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
50	9, 40, 40, 49	syl3anc 1249	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
51	48, 50	eqtrd 2222	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( +g ` W ) .1. ) )
52	37, 45, 51	3eqtr2d 2228	. . . . . . . 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 12972	. . . . . . . . 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 1251	. . . . . . . 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 2226	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .x. ( 0g ` W ) ) = .1. )
58	21, 25, 57	3eqtr3rd 2231	. . . . 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 2404	. 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 980   = wceq 1364   e. wcel 2160   =/= wne 2360   ` cfv 5231  (class class class)co 5891   Basecbs 12480   +g cplusg 12555  Scalarcsca 12558   .scvsca 12559   0gc0g 12727   +fcplusf 12795   Grpcgrp 12911   1rcur 13274   LModclmod 13564   .sfcscaf 13565

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-pre-ltirr 7941  ax-pre-ltadd 7945

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-pnf 8012  df-mnf 8013  df-ltxr 8015  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-5 8999  df-6 9000  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-plusg 12568  df-mulr 12569  df-sca 12571  df-vsca 12572  df-0g 12729  df-plusf 12797  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915  df-mgp 13236  df-ur 13275  df-ring 13313  df-lmod 13566  df-scaf 13567

This theorem is referenced by: (None)