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Theorem lmodfopne 13421
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 13420 . . . . 5 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))
9 simpll 527 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ π‘Š ∈ LMod)
10 simpl 109 . . . . . . . . 9 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 0 ∈ 𝑉)
1110adantl 277 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 0 ∈ 𝑉)
12 eqid 2177 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
133, 12lmod0vcl 13412 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
1413ad2antrr 488 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
15 eqid 2177 . . . . . . . . . 10 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
163, 15, 2plusfvalg 12787 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)))
1716eqcomd 2183 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 0 ∈ 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
189, 11, 14, 17syl3anc 1238 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 + (0gβ€˜π‘Š)))
19 oveq 5883 . . . . . . . 8 ( + = Β· β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
2019ad2antlr 489 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 + (0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
2118, 20eqtrd 2210 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 0 Β· (0gβ€˜π‘Š)))
22 lmodgrp 13389 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
2322adantr 276 . . . . . . 7 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ π‘Š ∈ Grp)
243, 15, 12grprid 12912 . . . . . . 7 ((π‘Š ∈ Grp ∧ 0 ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
2523, 10, 24syl2an 289 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 0 )
264, 5, 6lmod0cl 13409 . . . . . . . . 9 (π‘Š ∈ LMod β†’ 0 ∈ 𝐾)
2726ad2antrr 488 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 0 ∈ 𝐾)
28 eqid 2177 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
293, 4, 5, 1, 28scafvalg 13402 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 0 ∈ 𝐾 ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ( 0 Β· (0gβ€˜π‘Š)) = ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)))
309, 27, 14, 29syl3anc 1238 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)))
3126ancli 323 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
3231ad2antrr 488 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (π‘Š ∈ LMod ∧ 0 ∈ 𝐾))
334, 28, 5, 12lmodvs0 13417 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 0 ∈ 𝐾) β†’ ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
3432, 33syl 14 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 ( ·𝑠 β€˜π‘Š)(0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
35 simpr 110 . . . . . . . . . 10 (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ 1 ∈ 𝑉)
363, 15, 12grprid 12912 . . . . . . . . . 10 ((π‘Š ∈ Grp ∧ 1 ∈ 𝑉) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
3723, 35, 36syl2an 289 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 1 )
384, 5, 7lmod1cl 13410 . . . . . . . . . . . 12 (π‘Š ∈ LMod β†’ 1 ∈ 𝐾)
3938ad2antrr 488 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 ∈ 𝐾)
4035adantl 277 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 ∈ 𝑉)
413, 4, 5, 1, 28scafvalg 13402 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 1 ∈ 𝐾 ∧ 1 ∈ 𝑉) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
429, 39, 40, 41syl3anc 1238 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 ( ·𝑠 β€˜π‘Š) 1 ))
433, 4, 28, 7lmodvs1 13411 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 1 ∈ 𝑉) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4443ad2ant2rl 511 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 ( ·𝑠 β€˜π‘Š) 1 ) = 1 )
4542, 44eqtrd 2210 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = 1 )
46 oveq 5883 . . . . . . . . . . . 12 ( + = Β· β†’ ( 1 + 1 ) = ( 1 Β· 1 ))
4746eqcomd 2183 . . . . . . . . . . 11 ( + = Β· β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
4847ad2antlr 489 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 + 1 ))
493, 15, 2plusfvalg 12787 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 1 ∈ 𝑉 ∧ 1 ∈ 𝑉) β†’ ( 1 + 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
509, 40, 40, 49syl3anc 1238 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 + 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5148, 50eqtrd 2210 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 Β· 1 ) = ( 1 (+gβ€˜π‘Š) 1 ))
5237, 45, 513eqtr2d 2216 . . . . . . . 8 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 1 (+gβ€˜π‘Š)(0gβ€˜π‘Š)) = ( 1 (+gβ€˜π‘Š) 1 ))
5322ad2antrr 488 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ π‘Š ∈ Grp)
543, 15grplcan 12937 . . . . . . . . 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 ))
5553, 14, 40, 40, 54syl13anc 1240 . . . . . . . 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 ))
5652, 55mpbid 147 . . . . . . 7 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ (0gβ€˜π‘Š) = 1 )
5730, 34, 563eqtrd 2214 . . . . . 6 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ ( 0 Β· (0gβ€˜π‘Š)) = 1 )
5821, 25, 573eqtr3rd 2219 . . . . 5 (((π‘Š ∈ LMod ∧ + = Β· ) ∧ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) β†’ 1 = 0 )
598, 58mpdan 421 . . . 4 ((π‘Š ∈ LMod ∧ + = Β· ) β†’ 1 = 0 )
6059ex 115 . . 3 (π‘Š ∈ LMod β†’ ( + = Β· β†’ 1 = 0 ))
6160necon3d 2391 . 2 (π‘Š ∈ LMod β†’ ( 1 β‰  0 β†’ + β‰  Β· ))
6261imp 124 1 ((π‘Š ∈ LMod ∧ 1 β‰  0 ) β†’ + β‰  Β· )
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   β‰  wne 2347  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  Scalarcsca 12541   ·𝑠 cvsca 12542  0gc0g 12710  +𝑓cplusf 12777  Grpcgrp 12882  1rcur 13147  LModclmod 13382   Β·sf cscaf 13383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-sca 12554  df-vsca 12555  df-0g 12712  df-plusf 12779  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-mgp 13136  df-ur 13148  df-ring 13186  df-lmod 13384  df-scaf 13385
This theorem is referenced by: (None)
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