Theorem srascag 14427

Description: The set of scalars 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
srascag	⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))

Proof of Theorem srascag

Step	Hyp	Ref	Expression
1		srapart.ex	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
2		scaslid 13207	. . . . . . 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 13112	. . . . . . . 8 ⊢ Base Fn V
6	1	elexd 2813	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
7		funfvex 5649	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
8	7	funfni 5426	. . . . . . . 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 4224	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
12		ressex 13119	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑆 ∈ V) → (𝑊 ↾s 𝑆) ∈ V)
13	1, 11, 12	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V)
14		setsex 13085	. . . . 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		mulrslid 13186	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
17	16	slotex 13080	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → (.r‘𝑊) ∈ V)
18	1, 17	syl 14	. . . 4 ⊢ (𝜑 → (.r‘𝑊) ∈ V)
19		vscandxnscandx 13216	. . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
20	19	necomi 2485	. . . . 5 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx)
21		vscaslid 13217	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
22	21	simpri 113	. . . . 5 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
23	2, 20, 22	setsslnid 13105	. . . 4 ⊢ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
24	15, 18, 23	syl2anc 411	. . 3 ⊢ (𝜑 → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
25	22	a1i 9	. . . . 5 ⊢ (𝜑 → ( ·𝑠 ‘ndx) ∈ ℕ)
26		setsex 13085	. . . . 5 ⊢ (((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)
27	15, 25, 18, 26	syl3anc 1271	. . . 4 ⊢ (𝜑 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
28		slotsdifipndx 13229	. . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
29	28	simpri 113	. . . . 5 ⊢ (Scalar‘ndx) ≠ (·𝑖‘ndx)
30		ipslid 13225	. . . . . 6 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
31	30	simpri 113	. . . . 5 ⊢ (·𝑖‘ndx) ∈ ℕ
32	2, 29, 31	setsslnid 13105	. . . 4 ⊢ ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
33	27, 18, 32	syl2anc 411	. . 3 ⊢ (𝜑 → (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
34	24, 33	eqtrd 2262	. 2 ⊢ (𝜑 → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
35	2	setsslid 13104	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)))
36	1, 13, 35	syl2anc 411	. 2 ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)))
37		srapart.a	. . . 4 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
38		sraval 14422	. . . . 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)⟩))
39	6, 10, 38	syl2anc 411	. . . 4 ⊢ (𝜑 → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
40	37, 39	eqtrd 2262	. . 3 ⊢ (𝜑 → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
41	40	fveq2d 5636	. 2 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
42	34, 36, 41	3eqtr4d 2272	1 ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  Vcvv 2799   ⊆ wss 3197  ⟨cop 3669   Fn wfn 5316  ‘cfv 5321  (class class class)co 6010  ℕcn 9126  ndxcnx 13050   sSet csts 13051  Slot cslot 13052  Basecbs 13053   ↾_s cress 13054  ._rcmulr 13132  Scalarcsca 13134   ·_𝑠 cvsca 13135  ·_𝑖cip 13136  subringAlg csra 14418

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 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-mulr 13145  df-sca 13147  df-vsca 13148  df-ip 13149  df-sra 14420

This theorem is referenced by:  sralmod  14435  rlmscabas  14445