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Theorem lmodfopne 20502
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 20501 . . . . 5 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))
9 simpl 483 . . . . . . . 8 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 0 ∈ 𝑉)
10 eqid 2732 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
113, 10lmod0vcl 20493 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
1211adantr 481 . . . . . . . 8 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
13 eqid 2732 . . . . . . . . . 10 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
143, 13, 2plusfval 18564 . . . . . . . . 9 (( 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)))
1514eqcomd 2738 . . . . . . . 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 7411 . . . . . . . 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 2772 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
20 lmodgrp 20470 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
2120adantr 481 . . . . . . 7 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ π‘Š ∈ Grp)
223, 13, 10grprid 18849 . . . . . . 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 20490 . . . . . . . . . . 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 2732 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
293, 4, 5, 1, 28scafval 20483 . . . . . . . 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 20498 . . . . . . . 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 18849 . . . . . . . . . 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 20491 . . . . . . . . . . . 12 (π‘Š ∈ LMod β†’ 1 ∈ 𝐾)
4039adantr 481 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 ∈ 𝐾)
413, 4, 5, 1, 28scafval 20483 . . . . . . . . . . 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 20492 . . . . . . . . . . 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 2772 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = 1 )
46 oveq 7411 . . . . . . . . . . . 12 ( + = Β· β†’ ( 1 + 1 ) = ( 1 Β· 1 ))
4746eqcomd 2738 . . . . . . . . . . 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 18564 . . . . . . . . . . 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 2772 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5438, 45, 533eqtr2d 2778 . . . . . . . 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 18881 . . . . . . . . 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 2776 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = 1 )
6219, 23, 613eqtr3rd 2781 . . . . 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 2961 . 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 2940  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  +𝑓cplusf 18554  Grpcgrp 18815  1rcur 19998  LModclmod 20463   Β·sf cscaf 20464
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-plusf 18556  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-mgp 19982  df-ur 19999  df-ring 20051  df-lmod 20465  df-scaf 20466
This theorem is referenced by:  clmopfne  24603
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