Theorem sralemg 14586

Description: Lemma for srabaseg 14587 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 13271	. . . . . . 7 ⊢ Base Fn V
3	1	elexd 2827	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ V)
4		funfvex 5687	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
5	4	funfni 5458	. . . . . . 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 4250	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
9		ressex 13278	. . . . 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 2497	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
14		scaslid 13366	. . . . . 6 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
15	14	simpri 113	. . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ
16	11, 13, 15	setsslnid 13264	. . . 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 13244	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
20	1, 18, 10, 19	syl3anc 1274	. . . 4 ⊢ (𝜑 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
21		mulrslid 13345	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
22	21	slotex 13239	. . . . 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 2497	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
26		vscaslid 13376	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . 5 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
28	11, 25, 27	setsslnid 13264	. . . 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 13244	. . . . 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 1274	. . . 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 2497	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx)
35		ipslid 13384	. . . . . 6 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
36	35	simpri 113	. . . . 5 ⊢ (·𝑖‘ndx) ∈ ℕ
37	11, 34, 36	setsslnid 13264	. . . 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 2269	. 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 14585	. . . . 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 2265	. . 3 ⊢ (𝜑 → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
44	43	fveq2d 5674	. 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 2268	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  Vcvv 2813   ⊆ wss 3211  ⟨cop 3692   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  ℕcn 9237  ndxcnx 13209   sSet csts 13210  Slot cslot 13211  Basecbs 13212   ↾_s cress 13213  ._rcmulr 13291  Scalarcsca 13293   ·_𝑠 cvsca 13294  ·_𝑖cip 13295  subringAlg csra 14581

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-sra 14583

This theorem is referenced by:  srabaseg  14587  sraaddgg  14588  sramulrg  14589  sratsetg  14593  sradsg  14596