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Theorem asclrhm 19336
Description: The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
Hypotheses
Ref Expression
asclrhm.a 𝐴 = (algSc‘𝑊)
asclrhm.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
asclrhm (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))

Proof of Theorem asclrhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . 2 (Base‘𝐹) = (Base‘𝐹)
2 eqid 2621 . 2 (1r𝐹) = (1r𝐹)
3 eqid 2621 . 2 (1r𝑊) = (1r𝑊)
4 eqid 2621 . 2 (.r𝐹) = (.r𝐹)
5 eqid 2621 . 2 (.r𝑊) = (.r𝑊)
6 asclrhm.f . . . 4 𝐹 = (Scalar‘𝑊)
76assasca 19315 . . 3 (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
8 crngring 18552 . . 3 (𝐹 ∈ CRing → 𝐹 ∈ Ring)
97, 8syl 17 . 2 (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
10 assaring 19314 . 2 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
111, 2ringidcl 18562 . . . 4 (𝐹 ∈ Ring → (1r𝐹) ∈ (Base‘𝐹))
12 asclrhm.a . . . . 5 𝐴 = (algSc‘𝑊)
13 eqid 2621 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
1412, 6, 1, 13, 3asclval 19329 . . . 4 ((1r𝐹) ∈ (Base‘𝐹) → (𝐴‘(1r𝐹)) = ((1r𝐹)( ·𝑠𝑊)(1r𝑊)))
159, 11, 143syl 18 . . 3 (𝑊 ∈ AssAlg → (𝐴‘(1r𝐹)) = ((1r𝐹)( ·𝑠𝑊)(1r𝑊)))
16 assalmod 19313 . . . 4 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
17 eqid 2621 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
1817, 3ringidcl 18562 . . . . 5 (𝑊 ∈ Ring → (1r𝑊) ∈ (Base‘𝑊))
1910, 18syl 17 . . . 4 (𝑊 ∈ AssAlg → (1r𝑊) ∈ (Base‘𝑊))
2017, 6, 13, 2lmodvs1 18885 . . . 4 ((𝑊 ∈ LMod ∧ (1r𝑊) ∈ (Base‘𝑊)) → ((1r𝐹)( ·𝑠𝑊)(1r𝑊)) = (1r𝑊))
2116, 19, 20syl2anc 693 . . 3 (𝑊 ∈ AssAlg → ((1r𝐹)( ·𝑠𝑊)(1r𝑊)) = (1r𝑊))
2215, 21eqtrd 2655 . 2 (𝑊 ∈ AssAlg → (𝐴‘(1r𝐹)) = (1r𝑊))
2317, 5, 3ringlidm 18565 . . . . . . . 8 ((𝑊 ∈ Ring ∧ (1r𝑊) ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2410, 19, 23syl2anc 693 . . . . . . 7 (𝑊 ∈ AssAlg → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2524adantr 481 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2625oveq2d 6663 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊))) = (𝑦( ·𝑠𝑊)(1r𝑊)))
2726oveq2d 6663 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊)))) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
28 simpl 473 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ AssAlg)
29 simprl 794 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹))
3019adantr 481 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r𝑊) ∈ (Base‘𝑊))
3116adantr 481 . . . . . . 7 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod)
32 simprr 796 . . . . . . 7 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹))
3317, 6, 13, 1lmodvscl 18874 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r𝑊) ∈ (Base‘𝑊)) → (𝑦( ·𝑠𝑊)(1r𝑊)) ∈ (Base‘𝑊))
3431, 32, 30, 33syl3anc 1325 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠𝑊)(1r𝑊)) ∈ (Base‘𝑊))
3517, 6, 1, 13, 5assaass 19311 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ (1r𝑊) ∈ (Base‘𝑊) ∧ (𝑦( ·𝑠𝑊)(1r𝑊)) ∈ (Base‘𝑊))) → ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑥( ·𝑠𝑊)((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊)))))
3628, 29, 30, 34, 35syl13anc 1327 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑥( ·𝑠𝑊)((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊)))))
3717, 6, 1, 13, 5assaassr 19312 . . . . . . 7 ((𝑊 ∈ AssAlg ∧ (𝑦 ∈ (Base‘𝐹) ∧ (1r𝑊) ∈ (Base‘𝑊) ∧ (1r𝑊) ∈ (Base‘𝑊))) → ((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊))))
3828, 32, 30, 30, 37syl13anc 1327 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊))))
3938oveq2d 6663 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥( ·𝑠𝑊)((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊)))) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊)))))
4036, 39eqtrd 2655 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊)))))
4117, 6, 13, 1, 4lmodvsass 18882 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r𝑊) ∈ (Base‘𝑊))) → ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
4231, 29, 32, 30, 41syl13anc 1327 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
4327, 40, 423eqtr4rd 2666 . . 3 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)) = ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
441, 4ringcl 18555 . . . . . 6 ((𝐹 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
45443expb 1265 . . . . 5 ((𝐹 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
469, 45sylan 488 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
4712, 6, 1, 13, 3asclval 19329 . . . 4 ((𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹) → (𝐴‘(𝑥(.r𝐹)𝑦)) = ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)))
4846, 47syl 17 . . 3 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(.r𝐹)𝑦)) = ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)))
4912, 6, 1, 13, 3asclval 19329 . . . . 5 (𝑥 ∈ (Base‘𝐹) → (𝐴𝑥) = (𝑥( ·𝑠𝑊)(1r𝑊)))
5029, 49syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴𝑥) = (𝑥( ·𝑠𝑊)(1r𝑊)))
5112, 6, 1, 13, 3asclval 19329 . . . . 5 (𝑦 ∈ (Base‘𝐹) → (𝐴𝑦) = (𝑦( ·𝑠𝑊)(1r𝑊)))
5232, 51syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴𝑦) = (𝑦( ·𝑠𝑊)(1r𝑊)))
5350, 52oveq12d 6665 . . 3 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝐴𝑥)(.r𝑊)(𝐴𝑦)) = ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
5443, 48, 533eqtr4d 2665 . 2 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(.r𝐹)𝑦)) = ((𝐴𝑥)(.r𝑊)(𝐴𝑦)))
5512, 6, 10, 16asclghm 19332 . 2 (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 GrpHom 𝑊))
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 18722 1 (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  cfv 5886  (class class class)co 6647  Basecbs 15851  .rcmulr 15936  Scalarcsca 15938   ·𝑠 cvsca 15939  1rcur 18495  Ringcrg 18541  CRingccrg 18542   RingHom crh 18706  LModclmod 18857  AssAlgcasa 19303  algSccascl 19305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-plusg 15948  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-grp 17419  df-ghm 17652  df-mgp 18484  df-ur 18496  df-ring 18543  df-cring 18544  df-rnghom 18709  df-lmod 18859  df-assa 19306  df-ascl 19308
This theorem is referenced by:  mplind  19496  evlslem1  19509  mpfind  19530  pf1ind  19713  mat2pmatmul  20530  mat2pmatlin  20534
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