Theorem sralem 21061

Description: Lemma for srabase 21063 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 2992	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
4	1, 3	setsnid 17178	. . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩))
5		sralem.3	. . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
6	5	necomi 2992	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
7	1, 6	setsnid 17178	. . . 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 2992	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx)
10	1, 9	setsnid 17178	. . . 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 2760	. . 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 21060	. . . . . 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 592	. . . . 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 2768	. . . 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 6901	. . 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 2787	. 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
20	1	str0 17158	. . 3 ⊢ ∅ = (𝐸‘∅)
21		fvprc 6889	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅)
22	21	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅)
23		fv2prc 6942	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
24	12, 23	sylan9eqr 2790	. . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
25	24	fveq2d 6901	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅))
26	20, 22, 25	3eqtr4a 2794	. 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 1534   ∈ wcel 2099   ≠ wne 2937  Vcvv 3471   ⊆ wss 3947  ∅c0 4323  ⟨cop 4635  ‘cfv 6548  (class class class)co 7420   sSet csts 17132  Slot cslot 17150  ndxcnx 17162  Basecbs 17180   ↾_s cress 17209  ._rcmulr 17234  Scalarcsca 17236   ·_𝑠 cvsca 17237  ·_𝑖cip 17238  subringAlg csra 21056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-sets 17133  df-slot 17151  df-sra 21058

This theorem is referenced by:  srabase  21063  sraaddg  21065  sramulr  21067  sratset  21074  srads  21077  cchhllem  28710