Theorem sralem 21108

Description: Lemma for srabase 21109 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))
sralem.1	⊢ 𝐸 = Slot (𝐸‘ndx)
sralem.2	⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)
sralem.3	⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
sralem.4	⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)

Assertion

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

Proof of Theorem sralem

Step	Hyp	Ref	Expression
1		sralem.1	. . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		sralem.2	. . . . . 6 ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)
3	2	necomi 2982	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
4	1, 3	setsnid 17116	. . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩))
5		sralem.3	. . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
6	5	necomi 2982	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
7	1, 6	setsnid 17116	. . . 4 ⊢ (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩))
8		sralem.4	. . . . . 6 ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)
9	8	necomi 2982	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx)
10	1, 9	setsnid 17116	. . . 4 ⊢ (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)⟩))
11	4, 7, 10	3eqtri 2758	. . 3 ⊢ (𝐸‘𝑊) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
12		srapart.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
13	12	adantl 481	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
14		srapart.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
15		sraval 21107	. . . . . 6 ⊢ ((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)⟩))
16	14, 15	sylan2 593	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
17	13, 16	eqtrd 2766	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
18	17	fveq2d 6826	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
19	11, 18	eqtr4id 2785	. 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
20	1	str0 17097	. . 3 ⊢ ∅ = (𝐸‘∅)
21		fvprc 6814	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅)
22	21	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅)
23		fv2prc 6864	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
24	12, 23	sylan9eqr 2788	. . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
25	24	fveq2d 6826	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅))
26	20, 22, 25	3eqtr4a 2792	. 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
27	19, 26	pm2.61ian 811	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  ⟨cop 4582  ‘cfv 6481  (class class class)co 7346   sSet csts 17071  Slot cslot 17089  ndxcnx 17101  Basecbs 17117   ↾_s cress 17138  ._rcmulr 17159  Scalarcsca 17161   ·_𝑠 cvsca 17162  ·_𝑖cip 17163  subringAlg csra 21103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-sets 17072  df-slot 17090  df-sra 21105

This theorem is referenced by:  srabase  21109  sraaddg  21110  sramulr  21111  sratset  21115  srads  21117  cchhllem  28863