Theorem lmodfopne 14203

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 14202	. . . . 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 2207	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
13	3, 12	lmod0vcl 14194	. . . . . . . . 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 2207	. . . . . . . . . 10 |- ( +g ` W ) = ( +g ` W )
16	3, 15, 2	plusfvalg 13310	. . . . . . . . 9 |- ( ( W e. LMod /\ .0. e. V /\ ( 0g ` W ) e. V ) -> ( .0. .+ ( 0g ` W ) ) = ( .0. ( +g ` W ) ( 0g ` W ) ) )
17	16	eqcomd 2213	. . . . . . . 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 1250	. . . . . . 7 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .+ ( 0g ` W ) ) )
19		oveq 5973	. . . . . . . 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 2240	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. ( +g ` W ) ( 0g ` W ) ) = ( .0. .x. ( 0g ` W ) ) )
22		lmodgrp 14171	. . . . . . . 8 |- ( W e. LMod -> W e. Grp )
23	22	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ .+ = .x. ) -> W e. Grp )
24	3, 15, 12	grprid 13479	. . . . . . 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 14191	. . . . . . . . 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 2207	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
29	3, 4, 5, 1, 28	scafvalg 14184	. . . . . . . 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 1250	. . . . . . 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 14199	. . . . . . . 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 13479	. . . . . . . . . 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 14192	. . . . . . . . . . . 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 14184	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. K /\ .1. e. V ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
42	9, 39, 40, 41	syl3anc 1250	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( .s ` W ) .1. ) )
43	3, 4, 28, 7	lmodvs1 14193	. . . . . . . . . . 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 2240	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = .1. )
46		oveq 5973	. . . . . . . . . . . 12 |- ( .+ = .x. -> ( .1. .+ .1. ) = ( .1. .x. .1. ) )
47	46	eqcomd 2213	. . . . . . . . . . 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 13310	. . . . . . . . . . 11 |- ( ( W e. LMod /\ .1. e. V /\ .1. e. V ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
50	9, 40, 40, 49	syl3anc 1250	. . . . . . . . . 10 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .+ .1. ) = ( .1. ( +g ` W ) .1. ) )
51	48, 50	eqtrd 2240	. . . . . . . . 9 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .1. .x. .1. ) = ( .1. ( +g ` W ) .1. ) )
52	37, 45, 51	3eqtr2d 2246	. . . . . . . 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 13509	. . . . . . . . 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 1252	. . . . . . . 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 2244	. . . . . 6 |- ( ( ( W e. LMod /\ .+ = .x. ) /\ ( .0. e. V /\ .1. e. V ) ) -> ( .0. .x. ( 0g ` W ) ) = .1. )
58	21, 25, 57	3eqtr3rd 2249	. . . . 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 2422	. 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 981   = wceq 1373   e. wcel 2178   =/= wne 2378   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024  Scalarcsca 13027   .scvsca 13028   0gc0g 13203   +fcplusf 13300   Grpcgrp 13447   1rcur 13836   LModclmod 14164   .sfcscaf 14165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-sca 13040  df-vsca 13041  df-0g 13205  df-plusf 13302  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-mgp 13798  df-ur 13837  df-ring 13875  df-lmod 14166  df-scaf 14167

This theorem is referenced by: (None)