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Theorem asclrhm 19163
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 2610 . 2 (Base‘𝐹) = (Base‘𝐹)
2 eqid 2610 . 2 (1r𝐹) = (1r𝐹)
3 eqid 2610 . 2 (1r𝑊) = (1r𝑊)
4 eqid 2610 . 2 (.r𝐹) = (.r𝐹)
5 eqid 2610 . 2 (.r𝑊) = (.r𝑊)
6 asclrhm.f . . . 4 𝐹 = (Scalar‘𝑊)
76assasca 19142 . . 3 (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
8 crngring 18381 . . 3 (𝐹 ∈ CRing → 𝐹 ∈ Ring)
97, 8syl 17 . 2 (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
10 assaring 19141 . 2 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
111, 2ringidcl 18391 . . . 4 (𝐹 ∈ Ring → (1r𝐹) ∈ (Base‘𝐹))
12 asclrhm.a . . . . 5 𝐴 = (algSc‘𝑊)
13 eqid 2610 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
1412, 6, 1, 13, 3asclval 19156 . . . 4 ((1r𝐹) ∈ (Base‘𝐹) → (𝐴‘(1r𝐹)) = ((1r𝐹)( ·𝑠𝑊)(1r𝑊)))
159, 11, 143syl 18 . . 3 (𝑊 ∈ AssAlg → (𝐴‘(1r𝐹)) = ((1r𝐹)( ·𝑠𝑊)(1r𝑊)))
16 assalmod 19140 . . . 4 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
17 eqid 2610 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
1817, 3ringidcl 18391 . . . . 5 (𝑊 ∈ Ring → (1r𝑊) ∈ (Base‘𝑊))
1910, 18syl 17 . . . 4 (𝑊 ∈ AssAlg → (1r𝑊) ∈ (Base‘𝑊))
2017, 6, 13, 2lmodvs1 18714 . . . 4 ((𝑊 ∈ LMod ∧ (1r𝑊) ∈ (Base‘𝑊)) → ((1r𝐹)( ·𝑠𝑊)(1r𝑊)) = (1r𝑊))
2116, 19, 20syl2anc 691 . . 3 (𝑊 ∈ AssAlg → ((1r𝐹)( ·𝑠𝑊)(1r𝑊)) = (1r𝑊))
2215, 21eqtrd 2644 . 2 (𝑊 ∈ AssAlg → (𝐴‘(1r𝐹)) = (1r𝑊))
2317, 5, 3ringlidm 18394 . . . . . . . 8 ((𝑊 ∈ Ring ∧ (1r𝑊) ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2410, 19, 23syl2anc 691 . . . . . . 7 (𝑊 ∈ AssAlg → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2524adantr 480 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r𝑊)(.r𝑊)(1r𝑊)) = (1r𝑊))
2625oveq2d 6565 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊))) = (𝑦( ·𝑠𝑊)(1r𝑊)))
2726oveq2d 6565 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊)))) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
28 simpl 472 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ AssAlg)
29 simprl 790 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑥 ∈ (Base‘𝐹))
3019adantr 480 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r𝑊) ∈ (Base‘𝑊))
3116adantr 480 . . . . . . 7 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑊 ∈ LMod)
32 simprr 792 . . . . . . 7 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → 𝑦 ∈ (Base‘𝐹))
3317, 6, 13, 1lmodvscl 18703 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r𝑊) ∈ (Base‘𝑊)) → (𝑦( ·𝑠𝑊)(1r𝑊)) ∈ (Base‘𝑊))
3431, 32, 30, 33syl3anc 1318 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠𝑊)(1r𝑊)) ∈ (Base‘𝑊))
3517, 6, 1, 13, 5assaass 19138 . . . . . 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 1320 . . . . 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 19139 . . . . . . 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 1320 . . . . . 6 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r𝑊)(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))) = (𝑦( ·𝑠𝑊)((1r𝑊)(.r𝑊)(1r𝑊))))
3938oveq2d 6565 . . . . 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 2644 . . . 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 18711 . . . . 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 1320 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)) = (𝑥( ·𝑠𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
4327, 40, 423eqtr4rd 2655 . . 3 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r𝐹)𝑦)( ·𝑠𝑊)(1r𝑊)) = ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
441, 4ringcl 18384 . . . . . 6 ((𝐹 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
45443expb 1258 . . . . 5 ((𝐹 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
469, 45sylan 487 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r𝐹)𝑦) ∈ (Base‘𝐹))
4712, 6, 1, 13, 3asclval 19156 . . . 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 19156 . . . . 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 19156 . . . . 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 6567 . . 3 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝐴𝑥)(.r𝑊)(𝐴𝑦)) = ((𝑥( ·𝑠𝑊)(1r𝑊))(.r𝑊)(𝑦( ·𝑠𝑊)(1r𝑊))))
5443, 48, 533eqtr4d 2654 . 2 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(.r𝐹)𝑦)) = ((𝐴𝑥)(.r𝑊)(𝐴𝑦)))
5512, 6, 10, 16asclghm 19159 . 2 (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 GrpHom 𝑊))
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 18551 1 (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  1rcur 18324  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  LModclmod 18686  AssAlgcasa 19130  algSccascl 19132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-rnghom 18538  df-lmod 18688  df-assa 19133  df-ascl 19135
This theorem is referenced by:  mplind  19323  evlslem1  19336  mpfind  19357  pf1ind  19540  mat2pmatmul  20355  mat2pmatlin  20359
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