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Theorem lmodfopne 20345
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 Β· = ( Β·sf β€˜π‘Š)
lmodfopne.a + = (+π‘“β€˜π‘Š)
lmodfopne.v 𝑉 = (Baseβ€˜π‘Š)
lmodfopne.s 𝑆 = (Scalarβ€˜π‘Š)
lmodfopne.k 𝐾 = (Baseβ€˜π‘†)
lmodfopne.0 0 = (0gβ€˜π‘†)
lmodfopne.1 1 = (1rβ€˜π‘†)
Assertion
Ref Expression
lmodfopne ((π‘Š ∈ LMod ∧ 1 β‰  0 ) β†’ + β‰  Β· )

Proof of Theorem lmodfopne
StepHypRef Expression
1 lmodfopne.t . . . . . 6 Β· = ( Β·sf β€˜π‘Š)
2 lmodfopne.a . . . . . 6 + = (+π‘“β€˜π‘Š)
3 lmodfopne.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
4 lmodfopne.s . . . . . 6 𝑆 = (Scalarβ€˜π‘Š)
5 lmodfopne.k . . . . . 6 𝐾 = (Baseβ€˜π‘†)
6 lmodfopne.0 . . . . . 6 0 = (0gβ€˜π‘†)
7 lmodfopne.1 . . . . . 6 1 = (1rβ€˜π‘†)
81, 2, 3, 4, 5, 6, 7lmodfopnelem2 20344 . . . . 5 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))
9 simpl 483 . . . . . . . 8 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 0 ∈ 𝑉)
10 eqid 2736 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
113, 10lmod0vcl 20336 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
1211adantr 481 . . . . . . . 8 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
13 eqid 2736 . . . . . . . . . 10 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
143, 13, 2plusfval 18496 . . . . . . . . 9 (( 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)))
1514eqcomd 2742 . . . . . . . 8 (( 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
169, 12, 15syl2anr 597 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
17 oveq 7359 . . . . . . . 8 ( + = Β· β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
1817ad2antlr 725 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
1916, 18eqtrd 2776 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
20 lmodgrp 20314 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
2120adantr 481 . . . . . . 7 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ π‘Š ∈ Grp)
223, 13, 10grprid 18773 . . . . . . 7 ((π‘Š ∈ Grp ∧ 0 ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
2321, 9, 22syl2an 596 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
244, 5, 6lmod0cl 20333 . . . . . . . . . . 11 (π‘Š ∈ LMod β†’ 0 ∈ 𝐾)
2524, 11jca 512 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
2625adantr 481 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
2726adantr 481 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
28 eqid 2736 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
293, 4, 5, 1, 28scafval 20326 . . . . . . . 8 (( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 Β· (0gβ€˜π‘Š)) = ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)))
3027, 29syl 17 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)))
3124ancli 549 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
3231adantr 481 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
3332adantr 481 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
344, 28, 5, 10lmodvs0 20341 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 0 ∈ 𝐾) β†’ ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
3533, 34syl 17 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
36 simpr 485 . . . . . . . . . 10 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 1 ∈ 𝑉)
373, 13, 10grprid 18773 . . . . . . . . . 10 ((π‘Š ∈ Grp ∧ 1 ∈ 𝑉) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
3821, 36, 37syl2an 596 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
394, 5, 7lmod1cl 20334 . . . . . . . . . . . 12 (π‘Š ∈ LMod β†’ 1 ∈ 𝐾)
4039adantr 481 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 ∈ 𝐾)
413, 4, 5, 1, 28scafval 20326 . . . . . . . . . . 11 (( 1 ∈ 𝐾 ∧ 1 ∈ 𝑉) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
4240, 36, 41syl2an 596 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
433, 4, 28, 7lmodvs1 20335 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 1 ∈ 𝑉) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4443ad2ant2rl 747 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4542, 44eqtrd 2776 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = 1 )
46 oveq 7359 . . . . . . . . . . . 12 ( + = Β· β†’ ( 1 + 1 ) = ( 1 Β· 1 ))
4746eqcomd 2742 . . . . . . . . . . 11 ( + = Β· β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
4847ad2antlr 725 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
4936, 36jca 512 . . . . . . . . . . . 12 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ ( 1 ∈ 𝑉 ∧ 1 ∈ 𝑉))
5049adantl 482 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 ∈ 𝑉 ∧ 1 ∈ 𝑉))
513, 13, 2plusfval 18496 . . . . . . . . . . 11 (( 1 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ ( 1 + 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5250, 51syl 17 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 + 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5348, 52eqtrd 2776 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5438, 45, 533eqtr2d 2782 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 1 (+gβ€˜π‘Š) 1 ))
5521adantr 481 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ π‘Š ∈ Grp)
5612adantr 481 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
5736adantl 482 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 ∈ 𝑉)
583, 13grplcan 18800 . . . . . . . . 9 ((π‘Š ∈ Grp ∧ ((0gβ€˜π‘Š) ∈ 𝑉 ∧ 1 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 1 (+gβ€˜π‘Š) 1 ) ↔ (0gβ€˜π‘Š) = 1 ))
5955, 56, 57, 57, 58syl13anc 1372 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 1 (+gβ€˜π‘Š) 1 ) ↔ (0gβ€˜π‘Š) = 1 ))
6054, 59mpbid 231 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (0gβ€˜π‘Š) = 1 )
6130, 35, 603eqtrd 2780 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = 1 )
6219, 23, 613eqtr3rd 2785 . . . . 5 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 = 0 )
638, 62mpdan 685 . . . 4 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 = 0 )
6463ex 413 . . 3 (π‘Š ∈ LMod β†’ ( + = Β· β†’ 1 = 0 ))
6564necon3d 2962 . 2 (π‘Š ∈ LMod β†’ ( 1 β‰  0 β†’ + β‰  Β· ))
6665imp 407 1 ((π‘Š ∈ LMod ∧ 1 β‰  0 ) β†’ + β‰  Β· )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2941  β€˜cfv 6493  (class class class)co 7353  Basecbs 17075  +gcplusg 17125  Scalarcsca 17128   ·𝑠 cvsca 17129  0gc0g 17313  +𝑓cplusf 18486  Grpcgrp 18740  1rcur 19904  LModclmod 20307   Β·sf cscaf 20308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-plusg 17138  df-0g 17315  df-plusf 18488  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-minusg 18744  df-mgp 19888  df-ur 19905  df-ring 19952  df-lmod 20309  df-scaf 20310
This theorem is referenced by:  clmopfne  24443
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