Theorem sralemg 14072

Description: Lemma for srabaseg 14073 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)

Hypotheses

Ref	Expression
srapart.a	⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s	⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
srapart.ex	⊢ (𝜑 → 𝑊 ∈ 𝑋)
sralemg.1	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
sralem.2	⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)
sralem.3	⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
sralem.4	⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)

Assertion

Ref	Expression
sralemg	⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Proof of Theorem sralemg

Step	Hyp	Ref	Expression
1		srapart.ex	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
2		basfn 12763	. . . . . . 7 ⊢ Base Fn V
3	1	elexd 2776	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ V)
4		funfvex 5578	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
5	4	funfni 5361	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
6	2, 3, 5	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) ∈ V)
7		srapart.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
8	6, 7	ssexd 4174	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
9		ressex 12770	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑆 ∈ V) → (𝑊 ↾s 𝑆) ∈ V)
10	1, 8, 9	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V)
11		sralemg.1	. . . . 5 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
12		sralem.2	. . . . . 6 ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)
13	12	necomi 2452	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
14		scaslid 12857	. . . . . 6 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
15	14	simpri 113	. . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ
16	11, 13, 15	setsslnid 12757	. . . 4 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)))
17	1, 10, 16	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)))
18	15	a1i 9	. . . . 5 ⊢ (𝜑 → (Scalar‘ndx) ∈ ℕ)
19		setsex 12737	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
20	1, 18, 10, 19	syl3anc 1249	. . . 4 ⊢ (𝜑 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
21		mulrslid 12836	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
22	21	slotex 12732	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → (.r‘𝑊) ∈ V)
23	1, 22	syl 14	. . . 4 ⊢ (𝜑 → (.r‘𝑊) ∈ V)
24		sralem.3	. . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
25	24	necomi 2452	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
26		vscaslid 12867	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . 5 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
28	11, 25, 27	setsslnid 12757	. . . 4 ⊢ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
29	20, 23, 28	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)))
30	27	a1i 9	. . . . 5 ⊢ (𝜑 → ( ·𝑠 ‘ndx) ∈ ℕ)
31		setsex 12737	. . . . 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)
32	20, 30, 23, 31	syl3anc 1249	. . . 4 ⊢ (𝜑 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
33		sralem.4	. . . . . 6 ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)
34	33	necomi 2452	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx)
35		ipslid 12875	. . . . . 6 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
36	35	simpri 113	. . . . 5 ⊢ (·𝑖‘ndx) ∈ ℕ
37	11, 34, 36	setsslnid 12757	. . . 4 ⊢ ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V ∧ (.r‘𝑊) ∈ V) → (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
38	32, 23, 37	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
39	17, 29, 38	3eqtrd 2233	. 2 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
40		srapart.a	. . . 4 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
41		sraval 14071	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
42	1, 7, 41	syl2anc 411	. . . 4 ⊢ (𝜑 → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
43	40, 42	eqtrd 2229	. . 3 ⊢ (𝜑 → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
44	43	fveq2d 5565	. 2 ⊢ (𝜑 → (𝐸‘𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
45	39, 44	eqtr4d 2232	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  Vcvv 2763   ⊆ wss 3157  ⟨cop 3626   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925  ℕcn 9009  ndxcnx 12702   sSet csts 12703  Slot cslot 12704  Basecbs 12705   ↾_s cress 12706  ._rcmulr 12783  Scalarcsca 12785   ·_𝑠 cvsca 12786  ·_𝑖cip 12787  subringAlg csra 14067

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1re 7992  ax-addrcl 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-5 9071  df-6 9072  df-7 9073  df-8 9074  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-mulr 12796  df-sca 12798  df-vsca 12799  df-ip 12800  df-sra 14069

This theorem is referenced by:  srabaseg  14073  sraaddgg  14074  sramulrg  14075  sratsetg  14079  sradsg  14082