Theorem sralem 21140

Description: Lemma for srabase 21141 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 2987	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
4	1, 3	setsnid 17147	. . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩))
5		sralem.3	. . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
6	5	necomi 2987	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
7	1, 6	setsnid 17147	. . . 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 2987	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx)
10	1, 9	setsnid 17147	. . . 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 2764	. . 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 21139	. . . . . 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 594	. . . . 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 2772	. . . 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 6846	. . 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 2791	. 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
20	1	str0 17128	. . 3 ⊢ ∅ = (𝐸‘∅)
21		fvprc 6834	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅)
22	21	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅)
23		fv2prc 6884	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
24	12, 23	sylan9eqr 2794	. . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
25	24	fveq2d 6846	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅))
26	20, 22, 25	3eqtr4a 2798	. 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
27	19, 26	pm2.61ian 812	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368   sSet csts 17102  Slot cslot 17120  ndxcnx 17132  Basecbs 17148   ↾_s cress 17169  ._rcmulr 17190  Scalarcsca 17192   ·_𝑠 cvsca 17193  ·_𝑖cip 17194  subringAlg csra 21135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-sets 17103  df-slot 17121  df-sra 21137

This theorem is referenced by:  srabase  21141  sraaddg  21142  sramulr  21143  sratset  21147  srads  21149  cchhllem  28971