Theorem sravscag 14392

Description: The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)

Hypotheses

Ref	Expression
srapart.a	⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s	⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
srapart.ex	⊢ (𝜑 → 𝑊 ∈ 𝑋)

Assertion

Ref	Expression
sravscag	⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))

Proof of Theorem sravscag

Step	Hyp	Ref	Expression
1		srapart.ex	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
2		scaslid 13172	. . . . . . 7 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
3	2	simpri 113	. . . . . 6 ⊢ (Scalar‘ndx) ∈ ℕ
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → (Scalar‘ndx) ∈ ℕ)
5		basfn 13077	. . . . . . . 8 ⊢ Base Fn V
6	1	elexd 2813	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
7		funfvex 5640	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
8	7	funfni 5419	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
9	5, 6, 8	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑊) ∈ V)
10		srapart.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
11	9, 10	ssexd 4223	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
12		ressex 13084	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑆 ∈ V) → (𝑊 ↾s 𝑆) ∈ V)
13	1, 11, 12	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V)
14		setsex 13050	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
15	1, 4, 13, 14	syl3anc 1271	. . . 4 ⊢ (𝜑 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
16		vscaslid 13182	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
17	16	simpri 113	. . . . 5 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
18	17	a1i 9	. . . 4 ⊢ (𝜑 → ( ·𝑠 ‘ndx) ∈ ℕ)
19		mulrslid 13151	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
20	19	slotex 13045	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → (.r‘𝑊) ∈ V)
21	1, 20	syl 14	. . . 4 ⊢ (𝜑 → (.r‘𝑊) ∈ V)
22		setsex 13050	. . . 4 ⊢ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V ∧ ( ·𝑠 ‘ndx) ∈ ℕ ∧ (.r‘𝑊) ∈ V) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
23	15, 18, 21, 22	syl3anc 1271	. . 3 ⊢ (𝜑 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
24		slotsdifipndx 13194	. . . . 5 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
25	24	simpli 111	. . . 4 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx)
26		ipslid 13190	. . . . 5 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
27	26	simpri 113	. . . 4 ⊢ (·𝑖‘ndx) ∈ ℕ
28	16, 25, 27	setsslnid 13070	. . 3 ⊢ ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → ( ·𝑠 ‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = ( ·𝑠 ‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
29	23, 21, 28	syl2anc 411	. 2 ⊢ (𝜑 → ( ·𝑠 ‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = ( ·𝑠 ‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
30	16	setsslid 13069	. . 3 ⊢ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
31	15, 21, 30	syl2anc 411	. 2 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
32		srapart.a	. . . 4 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
33		sraval 14386	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
34	6, 10, 33	syl2anc 411	. . . 4 ⊢ (𝜑 → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
35	32, 34	eqtrd 2262	. . 3 ⊢ (𝜑 → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
36	35	fveq2d 5627	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
37	29, 31, 36	3eqtr4d 2272	1 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  Vcvv 2799   ⊆ wss 3197  ⟨cop 3669   Fn wfn 5309  ‘cfv 5314  (class class class)co 5994  ℕcn 9098  ndxcnx 13015   sSet csts 13016  Slot cslot 13017  Basecbs 13018   ↾_s cress 13019  ._rcmulr 13097  Scalarcsca 13099   ·_𝑠 cvsca 13100  ·_𝑖cip 13101  subringAlg csra 14382

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-mulr 13110  df-sca 13112  df-vsca 13113  df-ip 13114  df-sra 14384

This theorem is referenced by:  sralmod  14399  rlmvscag  14410