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Theorem lmodfopne 19595
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 19594 . . . . 5 ((𝑊 ∈ LMod ∧ + = · ) → ( 0𝑉1𝑉))
9 simpl 483 . . . . . . . 8 (( 0𝑉1𝑉) → 0𝑉)
10 eqid 2825 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
113, 10lmod0vcl 19586 . . . . . . . . 9 (𝑊 ∈ LMod → (0g𝑊) ∈ 𝑉)
1211adantr 481 . . . . . . . 8 ((𝑊 ∈ LMod ∧ + = · ) → (0g𝑊) ∈ 𝑉)
13 eqid 2825 . . . . . . . . . 10 (+g𝑊) = (+g𝑊)
143, 13, 2plusfval 17851 . . . . . . . . 9 (( 0𝑉 ∧ (0g𝑊) ∈ 𝑉) → ( 0 + (0g𝑊)) = ( 0 (+g𝑊)(0g𝑊)))
1514eqcomd 2831 . . . . . . . 8 (( 0𝑉 ∧ (0g𝑊) ∈ 𝑉) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 + (0g𝑊)))
169, 12, 15syl2anr 596 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 + (0g𝑊)))
17 oveq 7157 . . . . . . . 8 ( + = · → ( 0 + (0g𝑊)) = ( 0 · (0g𝑊)))
1817ad2antlr 723 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 + (0g𝑊)) = ( 0 · (0g𝑊)))
1916, 18eqtrd 2860 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = ( 0 · (0g𝑊)))
20 lmodgrp 19564 . . . . . . . 8 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2120adantr 481 . . . . . . 7 ((𝑊 ∈ LMod ∧ + = · ) → 𝑊 ∈ Grp)
223, 13, 10grprid 18067 . . . . . . 7 ((𝑊 ∈ Grp ∧ 0𝑉) → ( 0 (+g𝑊)(0g𝑊)) = 0 )
2321, 9, 22syl2an 595 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 (+g𝑊)(0g𝑊)) = 0 )
244, 5, 6lmod0cl 19583 . . . . . . . . . . 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 2825 . . . . . . . . 9 ( ·𝑠𝑊) = ( ·𝑠𝑊)
293, 4, 5, 1, 28scafval 19576 . . . . . . . 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 19591 . . . . . . . 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 18067 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ 1𝑉) → ( 1 (+g𝑊)(0g𝑊)) = 1 )
3821, 36, 37syl2an 595 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 (+g𝑊)(0g𝑊)) = 1 )
394, 5, 7lmod1cl 19584 . . . . . . . . . . . 12 (𝑊 ∈ LMod → 1𝐾)
4039adantr 481 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ + = · ) → 1𝐾)
413, 4, 5, 1, 28scafval 19576 . . . . . . . . . . 11 (( 1𝐾1𝑉) → ( 1 · 1 ) = ( 1 ( ·𝑠𝑊) 1 ))
4240, 36, 41syl2an 595 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = ( 1 ( ·𝑠𝑊) 1 ))
433, 4, 28, 7lmodvs1 19585 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 1𝑉) → ( 1 ( ·𝑠𝑊) 1 ) = 1 )
4443ad2ant2rl 745 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 ( ·𝑠𝑊) 1 ) = 1 )
4542, 44eqtrd 2860 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = 1 )
46 oveq 7157 . . . . . . . . . . . 12 ( + = · → ( 1 + 1 ) = ( 1 · 1 ))
4746eqcomd 2831 . . . . . . . . . . 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 17851 . . . . . . . . . . 11 (( 1𝑉1𝑉) → ( 1 + 1 ) = ( 1 (+g𝑊) 1 ))
5250, 51syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 + 1 ) = ( 1 (+g𝑊) 1 ))
5348, 52eqtrd 2860 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 1 · 1 ) = ( 1 (+g𝑊) 1 ))
5438, 45, 533eqtr2d 2866 . . . . . . . 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 18094 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ ((0g𝑊) ∈ 𝑉1𝑉1𝑉)) → (( 1 (+g𝑊)(0g𝑊)) = ( 1 (+g𝑊) 1 ) ↔ (0g𝑊) = 1 ))
5955, 56, 57, 57, 58syl13anc 1366 . . . . . . . 8 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (( 1 (+g𝑊)(0g𝑊)) = ( 1 (+g𝑊) 1 ) ↔ (0g𝑊) = 1 ))
6054, 59mpbid 233 . . . . . . 7 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → (0g𝑊) = 1 )
6130, 35, 603eqtrd 2864 . . . . . 6 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → ( 0 · (0g𝑊)) = 1 )
6219, 23, 613eqtr3rd 2869 . . . . 5 (((𝑊 ∈ LMod ∧ + = · ) ∧ ( 0𝑉1𝑉)) → 1 = 0 )
638, 62mpdan 683 . . . 4 ((𝑊 ∈ LMod ∧ + = · ) → 1 = 0 )
6463ex 413 . . 3 (𝑊 ∈ LMod → ( + = ·1 = 0 ))
6564necon3d 3041 . 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 1530  wcel 2107  wne 3020  cfv 6351  (class class class)co 7151  Basecbs 16476  +gcplusg 16558  Scalarcsca 16561   ·𝑠 cvsca 16562  0gc0g 16706  +𝑓cplusf 17842  Grpcgrp 18036  1rcur 19174  LModclmod 19557   ·sf cscaf 19558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16479  df-slot 16480  df-base 16482  df-sets 16483  df-plusg 16571  df-0g 16708  df-plusf 17844  df-mgm 17845  df-sgrp 17893  df-mnd 17904  df-grp 18039  df-minusg 18040  df-mgp 19163  df-ur 19175  df-ring 19222  df-lmod 19559  df-scaf 19560
This theorem is referenced by:  clmopfne  23618
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