Metamath Proof Explorer

Theorem sumdmdlem

Description: Lemma for sumdmdi . The span of vector C not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004) (New usage is discouraged.)

Ref Expression
Hypotheses sumdmdi.1 ⊢ 𝐴 ∈ Cℋ
sumdmdi.2 ⊢ 𝐵 ∈ Cℋ
Assertion sumdmdlem ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 sumdmdi.1 ⊢ 𝐴 ∈ Cℋ
2 sumdmdi.2 ⊢ 𝐵 ∈ Cℋ
3 elin ⊢ ( 𝑦 ∈ ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) ↔ ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 ∈ 𝐴 ) )
4 2 chshii ⊢ 𝐵 ∈ Sℋ
5 spansnsh ⊢ ( 𝐶 ∈ ℋ → ( span ‘ { 𝐶 } ) ∈ Sℋ )
6 shsel ⊢ ( ( 𝐵 ∈ Sℋ ∧ ( span ‘ { 𝐶 } ) ∈ Sℋ ) → ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ↔ ∃ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ ( span ‘ { 𝐶 } ) 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) )
7 4 5 6 sylancr ⊢ ( 𝐶 ∈ ℋ → ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ↔ ∃ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ ( span ‘ { 𝐶 } ) 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) )
8 1 cheli ⊢ ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ )
9 2 cheli ⊢ ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ )
10 elspansncl ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) → 𝑤 ∈ ℋ )
11 hvsubadd ⊢ ( ( 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 ↔ ( 𝑧 +ℎ 𝑤 ) = 𝑦 ) )
12 eqcom ⊢ ( ( 𝑧 +ℎ 𝑤 ) = 𝑦 ↔ 𝑦 = ( 𝑧 +ℎ 𝑤 ) )
13 11 12 bitrdi ⊢ ( ( 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 ↔ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) )
14 8 9 10 13 syl3an ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ ( 𝐶 ∈ ℋ ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 ↔ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) )
15 14 3expa ⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐶 ∈ ℋ ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 ↔ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) )
16 1 chshii ⊢ 𝐴 ∈ Sℋ
17 16 4 shsvsi ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 −ℎ 𝑧 ) ∈ ( 𝐴 +ℋ 𝐵 ) )
18 eleq1 ⊢ ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 → ( ( 𝑦 −ℎ 𝑧 ) ∈ ( 𝐴 +ℋ 𝐵 ) ↔ 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
19 17 18 syl5ibcom ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
20 19 adantr ⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐶 ∈ ℋ ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ) → ( ( 𝑦 −ℎ 𝑧 ) = 𝑤 → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
21 15 20 sylbird ⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐶 ∈ ℋ ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
22 21 exp32 ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐶 ∈ ℋ → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) ) )
23 22 com4r ⊢ ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐶 ∈ ℋ → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) ) )
24 23 imp31 ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝐶 ∈ ℋ ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
25 24 adantrr ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
26 16 4 shscli ⊢ ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ
27 elspansn5 ⊢ ( ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ → ( ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ∧ ( 𝑤 ∈ ( span ‘ { 𝐶 } ) ∧ 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → 𝑤 = 0ℎ ) )
28 26 27 ax-mp ⊢ ( ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ∧ ( 𝑤 ∈ ( span ‘ { 𝐶 } ) ∧ 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → 𝑤 = 0ℎ )
29 28 exp32 ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) → 𝑤 = 0ℎ ) ) )
30 29 adantl ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑤 ∈ ( 𝐴 +ℋ 𝐵 ) → 𝑤 = 0ℎ ) ) )
31 25 30 mpdd ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑤 = 0ℎ ) )
32 oveq2 ⊢ ( 𝑤 = 0ℎ → ( 𝑧 +ℎ 𝑤 ) = ( 𝑧 +ℎ 0ℎ ) )
33 ax-hvaddid ⊢ ( 𝑧 ∈ ℋ → ( 𝑧 +ℎ 0ℎ ) = 𝑧 )
34 32 33 sylan9eqr ⊢ ( ( 𝑧 ∈ ℋ ∧ 𝑤 = 0ℎ ) → ( 𝑧 +ℎ 𝑤 ) = 𝑧 )
35 9 34 sylan ⊢ ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 = 0ℎ ) → ( 𝑧 +ℎ 𝑤 ) = 𝑧 )
36 35 eqeq2d ⊢ ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 = 0ℎ ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ↔ 𝑦 = 𝑧 ) )
37 36 adantll ⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑤 = 0ℎ ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ↔ 𝑦 = 𝑧 ) )
38 37 biimpac ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑤 = 0ℎ ) ) → 𝑦 = 𝑧 )
39 eleq1 ⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) )
40 39 biimparc ⊢ ( ( 𝑧 ∈ 𝐵 ∧ 𝑦 = 𝑧 ) → 𝑦 ∈ 𝐵 )
41 elin ⊢ ( 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) )
42 41 biimpri ⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) )
43 42 ancoms ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) )
44 40 43 sylan2 ⊢ ( ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑦 = 𝑧 ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) )
45 44 expr ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 = 𝑧 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
46 45 ad2antrl ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑤 = 0ℎ ) ) → ( 𝑦 = 𝑧 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
47 38 46 mpd ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑤 = 0ℎ ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) )
48 47 expr ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑤 = 0ℎ → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
49 48 a1d ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑤 = 0ℎ → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) )
50 49 adantr ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑤 = 0ℎ → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) )
51 31 50 mpdd ⊢ ( ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
52 51 ex ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) )
53 52 com23 ⊢ ( ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) )
54 53 exp32 ⊢ ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → ( 𝑦 ∈ 𝐴 → ( 𝑧 ∈ 𝐵 → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) ) )
55 54 com4l ⊢ ( 𝑦 ∈ 𝐴 → ( 𝑧 ∈ 𝐵 → ( 𝑤 ∈ ( span ‘ { 𝐶 } ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) ) )
56 55 imp4c ⊢ ( 𝑦 ∈ 𝐴 → ( ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) → ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) )
57 56 exp4a ⊢ ( 𝑦 ∈ 𝐴 → ( ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) → ( 𝐶 ∈ ℋ → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
58 57 com23 ⊢ ( 𝑦 ∈ 𝐴 → ( 𝐶 ∈ ℋ → ( ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
59 58 com4l ⊢ ( 𝐶 ∈ ℋ → ( ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 = ( 𝑧 +ℎ 𝑤 ) ) → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
60 59 expd ⊢ ( 𝐶 ∈ ℋ → ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ ( span ‘ { 𝐶 } ) ) → ( 𝑦 = ( 𝑧 +ℎ 𝑤 ) → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) ) )
61 60 rexlimdvv ⊢ ( 𝐶 ∈ ℋ → ( ∃ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ ( span ‘ { 𝐶 } ) 𝑦 = ( 𝑧 +ℎ 𝑤 ) → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
62 7 61 sylbid ⊢ ( 𝐶 ∈ ℋ → ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
63 62 com23 ⊢ ( 𝐶 ∈ ℋ → ( ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) → ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) ) ) )
64 63 imp4b ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( ( 𝑦 ∈ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
65 3 64 biimtrid ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( 𝑦 ∈ ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐴 ) ) )
66 65 ssrdv ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) ⊆ ( 𝐵 ∩ 𝐴 ) )
67 shsub1 ⊢ ( ( 𝐵 ∈ Sℋ ∧ ( span ‘ { 𝐶 } ) ∈ Sℋ ) → 𝐵 ⊆ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) )
68 4 5 67 sylancr ⊢ ( 𝐶 ∈ ℋ → 𝐵 ⊆ ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) )
69 68 ssrind ⊢ ( 𝐶 ∈ ℋ → ( 𝐵 ∩ 𝐴 ) ⊆ ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) )
70 69 adantr ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( 𝐵 ∩ 𝐴 ) ⊆ ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) )
71 66 70 eqssd ⊢ ( ( 𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( ( 𝐵 +ℋ ( span ‘ { 𝐶 } ) ) ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) )