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Theorem lmodfopne 20510
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 20509 . . . . 5 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))
9 simpl 484 . . . . . . . 8 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 0 ∈ 𝑉)
10 eqid 2733 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
113, 10lmod0vcl 20501 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
1211adantr 482 . . . . . . . 8 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
13 eqid 2733 . . . . . . . . . 10 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
143, 13, 2plusfval 18568 . . . . . . . . 9 (( 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)))
1514eqcomd 2739 . . . . . . . 8 (( 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
169, 12, 15syl2anr 598 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
17 oveq 7415 . . . . . . . 8 ( + = Β· β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
1817ad2antlr 726 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
1916, 18eqtrd 2773 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
20 lmodgrp 20478 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
2120adantr 482 . . . . . . 7 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ π‘Š ∈ Grp)
223, 13, 10grprid 18853 . . . . . . 7 ((π‘Š ∈ Grp ∧ 0 ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
2321, 9, 22syl2an 597 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
244, 5, 6lmod0cl 20498 . . . . . . . . . . 11 (π‘Š ∈ LMod β†’ 0 ∈ 𝐾)
2524, 11jca 513 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
2625adantr 482 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
2726adantr 482 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉))
28 eqid 2733 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
293, 4, 5, 1, 28scafval 20491 . . . . . . . 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 550 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
3231adantr 482 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
3332adantr 482 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
344, 28, 5, 10lmodvs0 20506 . . . . . . . 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 486 . . . . . . . . . 10 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 1 ∈ 𝑉)
373, 13, 10grprid 18853 . . . . . . . . . 10 ((π‘Š ∈ Grp ∧ 1 ∈ 𝑉) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
3821, 36, 37syl2an 597 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
394, 5, 7lmod1cl 20499 . . . . . . . . . . . 12 (π‘Š ∈ LMod β†’ 1 ∈ 𝐾)
4039adantr 482 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 ∈ 𝐾)
413, 4, 5, 1, 28scafval 20491 . . . . . . . . . . 11 (( 1 ∈ 𝐾 ∧ 1 ∈ 𝑉) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
4240, 36, 41syl2an 597 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
433, 4, 28, 7lmodvs1 20500 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 1 ∈ 𝑉) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4443ad2ant2rl 748 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4542, 44eqtrd 2773 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = 1 )
46 oveq 7415 . . . . . . . . . . . 12 ( + = Β· β†’ ( 1 + 1 ) = ( 1 Β· 1 ))
4746eqcomd 2739 . . . . . . . . . . 11 ( + = Β· β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
4847ad2antlr 726 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
4936, 36jca 513 . . . . . . . . . . . 12 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ ( 1 ∈ 𝑉 ∧ 1 ∈ 𝑉))
5049adantl 483 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 ∈ 𝑉 ∧ 1 ∈ 𝑉))
513, 13, 2plusfval 18568 . . . . . . . . . . 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 2773 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5438, 45, 533eqtr2d 2779 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 1 (+gβ€˜π‘Š) 1 ))
5521adantr 482 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ π‘Š ∈ Grp)
5612adantr 482 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
5736adantl 483 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 ∈ 𝑉)
583, 13grplcan 18885 . . . . . . . . 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 1373 . . . . . . . 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 2777 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = 1 )
6219, 23, 613eqtr3rd 2782 . . . . 5 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 = 0 )
638, 62mpdan 686 . . . 4 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 = 0 )
6463ex 414 . . 3 (π‘Š ∈ LMod β†’ ( + = Β· β†’ 1 = 0 ))
6564necon3d 2962 . 2 (π‘Š ∈ LMod β†’ ( 1 β‰  0 β†’ + β‰  Β· ))
6665imp 408 1 ((π‘Š ∈ LMod ∧ 1 β‰  0 ) β†’ + β‰  Β· )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  +𝑓cplusf 18558  Grpcgrp 18819  1rcur 20004  LModclmod 20471   Β·sf cscaf 20472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-mgp 19988  df-ur 20005  df-ring 20058  df-lmod 20473  df-scaf 20474
This theorem is referenced by:  clmopfne  24612
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