Theorem asclrhm 19342

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2622	. 2 ⊢ (1r‘𝐹) = (1r‘𝐹)
3		eqid 2622	. 2 ⊢ (1r‘𝑊) = (1r‘𝑊)
4		eqid 2622	. 2 ⊢ (.r‘𝐹) = (.r‘𝐹)
5		eqid 2622	. 2 ⊢ (.r‘𝑊) = (.r‘𝑊)
6		asclrhm.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
7	6	assasca 19321	. . 3 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
8		crngring 18558	. . 3 ⊢ (𝐹 ∈ CRing → 𝐹 ∈ Ring)
9	7, 8	syl 17	. 2 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
10		assaring 19320	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
11	1, 2	ringidcl 18568	. . . 4 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹))
12		asclrhm.a	. . . . 5 ⊢ 𝐴 = (algSc‘𝑊)
13		eqid 2622	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
14	12, 6, 1, 13, 3	asclval 19335	. . . 4 ⊢ ((1r‘𝐹) ∈ (Base‘𝐹) → (𝐴‘(1r‘𝐹)) = ((1r‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)))
15	9, 11, 14	3syl 18	. . 3 ⊢ (𝑊 ∈ AssAlg → (𝐴‘(1r‘𝐹)) = ((1r‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)))
16		assalmod 19319	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
17		eqid 2622	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
18	17, 3	ringidcl 18568	. . . . 5 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊))
19	10, 18	syl 17	. . . 4 ⊢ (𝑊 ∈ AssAlg → (1r‘𝑊) ∈ (Base‘𝑊))
20	17, 6, 13, 2	lmodvs1 18891	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (1r‘𝑊) ∈ (Base‘𝑊)) → ((1r‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (1r‘𝑊))
21	16, 19, 20	syl2anc 693	. . 3 ⊢ (𝑊 ∈ AssAlg → ((1r‘𝐹)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (1r‘𝑊))
22	15, 21	eqtrd 2656	. 2 ⊢ (𝑊 ∈ AssAlg → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
23	17, 5, 3	ringlidm 18571	. . . . . . . 8 ⊢ ((𝑊 ∈ Ring ∧ (1r‘𝑊) ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)) = (1r‘𝑊))
24	10, 19, 23	syl2anc 693	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → ((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)) = (1r‘𝑊))
25	24	adantr 481	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)) = (1r‘𝑊))
26	25	oveq2d 6666	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊))) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))
27	26	oveq2d 6666	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)))) = (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))))
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))
30	19	adantr 481	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (1r‘𝑊) ∈ (Base‘𝑊))
31	16	adantr 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))
33	17, 6, 13, 1	lmodvscl 18880	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r‘𝑊) ∈ (Base‘𝑊)) → (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ (Base‘𝑊))
34	31, 32, 30, 33	syl3anc 1326	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ (Base‘𝑊))
35	17, 6, 1, 13, 5	assaass 19317	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ (1r‘𝑊) ∈ (Base‘𝑊) ∧ (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)) ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑥( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))))
36	28, 29, 30, 34, 35	syl13anc 1328	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑥( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))))
37	17, 6, 1, 13, 5	assaassr 19318	. . . . . . 7 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑦 ∈ (Base‘𝐹) ∧ (1r‘𝑊) ∈ (Base‘𝑊) ∧ (1r‘𝑊) ∈ (Base‘𝑊))) → ((1r‘𝑊)(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊))))
38	28, 32, 30, 30, 37	syl13anc 1328	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((1r‘𝑊)(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊))))
39	38	oveq2d 6666	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))) = (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)))))
40	36, 39	eqtrd 2656	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))) = (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)((1r‘𝑊)(.r‘𝑊)(1r‘𝑊)))))
41	17, 6, 13, 1, 4	lmodvsass 18888	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹) ∧ (1r‘𝑊) ∈ (Base‘𝑊))) → ((𝑥(.r‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))))
42	31, 29, 32, 30, 41	syl13anc 1328	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑥( ·𝑠 ‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))))
43	27, 40, 42	3eqtr4rd 2667	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(.r‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))))
44	1, 4	ringcl 18561	. . . . . 6 ⊢ ((𝐹 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹)) → (𝑥(.r‘𝐹)𝑦) ∈ (Base‘𝐹))
45	44	3expb 1266	. . . . 5 ⊢ ((𝐹 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r‘𝐹)𝑦) ∈ (Base‘𝐹))
46	9, 45	sylan 488	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝑥(.r‘𝐹)𝑦) ∈ (Base‘𝐹))
47	12, 6, 1, 13, 3	asclval 19335	. . . 4 ⊢ ((𝑥(.r‘𝐹)𝑦) ∈ (Base‘𝐹) → (𝐴‘(𝑥(.r‘𝐹)𝑦)) = ((𝑥(.r‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)))
48	46, 47	syl 17	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(.r‘𝐹)𝑦)) = ((𝑥(.r‘𝐹)𝑦)( ·𝑠 ‘𝑊)(1r‘𝑊)))
49	12, 6, 1, 13, 3	asclval 19335	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐹) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)))
50	29, 49	syl 17	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘𝑥) = (𝑥( ·𝑠 ‘𝑊)(1r‘𝑊)))
51	12, 6, 1, 13, 3	asclval 19335	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝐹) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))
52	32, 51	syl 17	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘𝑦) = (𝑦( ·𝑠 ‘𝑊)(1r‘𝑊)))
53	50, 52	oveq12d 6668	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝐴‘𝑥)(.r‘𝑊)(𝐴‘𝑦)) = ((𝑥( ·𝑠 ‘𝑊)(1r‘𝑊))(.r‘𝑊)(𝑦( ·𝑠 ‘𝑊)(1r‘𝑊))))
54	43, 48, 53	3eqtr4d 2666	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → (𝐴‘(𝑥(.r‘𝐹)𝑦)) = ((𝐴‘𝑥)(.r‘𝑊)(𝐴‘𝑦)))
55	12, 6, 10, 16	asclghm 19338	. 2 ⊢ (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 GrpHom 𝑊))
56	1, 2, 3, 4, 5, 9, 10, 22, 54, 55	isrhm2d 18728	1 ⊢ (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  1_rcur 18501  Ringcrg 18547  CRingccrg 18548   RingHom crh 18712  LModclmod 18863  AssAlgcasa 19309  algSccascl 19311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-rnghom 18715  df-lmod 18865  df-assa 19312  df-ascl 19314

This theorem is referenced by:  mplind  19502  evlslem1  19515  mpfind  19536  pf1ind  19719  mat2pmatmul  20536  mat2pmatlin  20540