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Theorem lmodfopne 19601
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 · = ( ·sf𝑊)
lmodfopne.a + = (+𝑓𝑊)
lmodfopne.v 𝑉 = (Base‘𝑊)
lmodfopne.s 𝑆 = (Scalar‘𝑊)
lmodfopne.k 𝐾 = (Base‘𝑆)
lmodfopne.0 0 = (0g𝑆)
lmodfopne.1 1 = (1r𝑆)
Assertion
Ref Expression
lmodfopne ((𝑊 ∈ LMod ∧ 10 ) → +· )

Proof of Theorem lmodfopne
StepHypRef Expression
1 lmodfopne.t . . . . . 6 · = ( ·sf𝑊)
2 lmodfopne.a . . . . . 6 + = (+𝑓𝑊)
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 = (0g𝑆)
7 lmodfopne.1 . . . . . 6 1 = (1r𝑆)
81, 2, 3, 4, 5, 6, 7lmodfopnelem2 19600 . . . . 5 ((𝑊 ∈ LMod ∧ + = · ) → ( 0𝑉1𝑉))
9 simpl 483 . . . . . . . 8 (( 0𝑉1𝑉) → 0𝑉)
10 eqid 2818 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
113, 10lmod0vcl 19592 . . . . . . . . 9 (𝑊 ∈ LMod → (0g𝑊) ∈ 𝑉)
1211adantr 481 . . . . . . . 8 ((𝑊 ∈ LMod ∧ + = · ) → (0g𝑊) ∈ 𝑉)
13 eqid 2818 . . . . . . . . . 10 (+g𝑊) = (+g𝑊)
143, 13, 2plusfval 17847 . . . . . . . . 9 (( 0𝑉 ∧ (0g𝑊) ∈ 𝑉) → ( 0 + (0g𝑊)) = ( 0 (+g𝑊)(0g𝑊)))
1514eqcomd 2824 . . . . . . . 8 (( 0𝑉 ∧ (0g𝑊) ∈ 𝑉) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 + (0g𝑊)))
169, 12, 15syl2anr 596 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 + (0g𝑊)))
17 oveq 7151 . . . . . . . 8 ( + = · → ( 0 + (0g𝑊)) = ( 0 · (0g𝑊)))
1817ad2antlr 723 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 + (0g𝑊)) = ( 0 · (0g𝑊)))
1916, 18eqtrd 2853 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 · (0g𝑊)))
20 lmodgrp 19570 . . . . . . . 8 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2120adantr 481 . . . . . . 7 ((𝑊 ∈ LMod ∧ + = · ) → 𝑊 ∈ Grp)
223, 13, 10grprid 18072 . . . . . . 7 ((𝑊 ∈ Grp ∧ 0𝑉) → ( 0 (+g𝑊)(0g𝑊)) = 0 )
2321, 9, 22syl2an 595 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = 0 )
244, 5, 6lmod0cl 19589 . . . . . . . . . . 11 (𝑊 ∈ LMod → 0𝐾)
2524, 11jca 512 . . . . . . . . . 10 (𝑊 ∈ LMod → ( 0𝐾 ∧ (0g𝑊) ∈ 𝑉))
2625adantr 481 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ + = · ) → ( 0𝐾 ∧ (0g𝑊) ∈ 𝑉))
2726adantr 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0𝐾 ∧ (0g𝑊) ∈ 𝑉))
28 eqid 2818 . . . . . . . . 9 ( ·𝑠𝑊) = ( ·𝑠𝑊)
293, 4, 5, 1, 28scafval 19582 . . . . . . . 8 (( 0𝐾 ∧ (0g𝑊) ∈ 𝑉) → ( 0 · (0g𝑊)) = ( 0 ( ·𝑠𝑊)(0g𝑊)))
3027, 29syl 17 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 · (0g𝑊)) = ( 0 ( ·𝑠𝑊)(0g𝑊)))
3124ancli 549 . . . . . . . . . 10 (𝑊 ∈ LMod → (𝑊 ∈ LMod ∧ 0𝐾))
3231adantr 481 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ + = · ) → (𝑊 ∈ LMod ∧ 0𝐾))
3332adantr 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (𝑊 ∈ LMod ∧ 0𝐾))
344, 28, 5, 10lmodvs0 19597 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 0𝐾) → ( 0 ( ·𝑠𝑊)(0g𝑊)) = (0g𝑊))
3533, 34syl 17 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 ( ·𝑠𝑊)(0g𝑊)) = (0g𝑊))
36 simpr 485 . . . . . . . . . 10 (( 0𝑉1𝑉) → 1𝑉)
373, 13, 10grprid 18072 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ 1𝑉) → ( 1 (+g𝑊)(0g𝑊)) = 1 )
3821, 36, 37syl2an 595 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 (+g𝑊)(0g𝑊)) = 1 )
394, 5, 7lmod1cl 19590 . . . . . . . . . . . 12 (𝑊 ∈ LMod → 1𝐾)
4039adantr 481 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ + = · ) → 1𝐾)
413, 4, 5, 1, 28scafval 19582 . . . . . . . . . . 11 (( 1𝐾1𝑉) → ( 1 · 1 ) = ( 1 ( ·𝑠𝑊) 1 ))
4240, 36, 41syl2an 595 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = ( 1 ( ·𝑠𝑊) 1 ))
433, 4, 28, 7lmodvs1 19591 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 1𝑉) → ( 1 ( ·𝑠𝑊) 1 ) = 1 )
4443ad2ant2rl 745 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 ( ·𝑠𝑊) 1 ) = 1 )
4542, 44eqtrd 2853 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = 1 )
46 oveq 7151 . . . . . . . . . . . 12 ( + = · → ( 1 + 1 ) = ( 1 · 1 ))
4746eqcomd 2824 . . . . . . . . . . 11 ( + = · → ( 1 · 1 ) = ( 1 + 1 ))
4847ad2antlr 723 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = ( 1 + 1 ))
4936, 36jca 512 . . . . . . . . . . . 12 (( 0𝑉1𝑉) → ( 1𝑉1𝑉))
5049adantl 482 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1𝑉1𝑉))
513, 13, 2plusfval 17847 . . . . . . . . . . 11 (( 1𝑉1𝑉) → ( 1 + 1 ) = ( 1 (+g𝑊) 1 ))
5250, 51syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 + 1 ) = ( 1 (+g𝑊) 1 ))
5348, 52eqtrd 2853 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = ( 1 (+g𝑊) 1 ))
5438, 45, 533eqtr2d 2859 . . . . . . . 8 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 (+g𝑊)(0g𝑊)) = ( 1 (+g𝑊) 1 ))
5521adantr 481 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → 𝑊 ∈ Grp)
5612adantr 481 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (0g𝑊) ∈ 𝑉)
5736adantl 482 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → 1𝑉)
583, 13grplcan 18099 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ ((0g𝑊) ∈ 𝑉1𝑉1𝑉)) → (( 1 (+g𝑊)(0g𝑊)) = ( 1 (+g𝑊) 1 ) ↔ (0g𝑊) = 1 ))
5955, 56, 57, 57, 58syl13anc 1364 . . . . . . . 8 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (( 1 (+g𝑊)(0g𝑊)) = ( 1 (+g𝑊) 1 ) ↔ (0g𝑊) = 1 ))
6054, 59mpbid 233 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (0g𝑊) = 1 )
6130, 35, 603eqtrd 2857 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 · (0g𝑊)) = 1 )
6219, 23, 613eqtr3rd 2862 . . . . 5 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → 1 = 0 )
638, 62mpdan 683 . . . 4 ((𝑊 ∈ LMod ∧ + = · ) → 1 = 0 )
6463ex 413 . . 3 (𝑊 ∈ LMod → ( + = ·1 = 0 ))
6564necon3d 3034 . 2 (𝑊 ∈ LMod → ( 10+· ))
6665imp 407 1 ((𝑊 ∈ LMod ∧ 10 ) → +· )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  +𝑓cplusf 17837  Grpcgrp 18041  1rcur 19180  LModclmod 19563   ·sf cscaf 19564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-plusg 16566  df-0g 16703  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-mgp 19169  df-ur 19181  df-ring 19228  df-lmod 19565  df-scaf 19566
This theorem is referenced by:  clmopfne  23627
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