Metamath Proof Explorer


Theorem cdleme48d

Description: TODO: fix comment. (Contributed by NM, 8-Apr-2013)

Ref Expression
Hypotheses cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef46g.l = ( le ‘ 𝐾 )
cdlemef46g.j = ( join ‘ 𝐾 )
cdlemef46g.m = ( meet ‘ 𝐾 )
cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
cdlemef46.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdleme48d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemef46g.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef46g.l = ( le ‘ 𝐾 )
3 cdlemef46g.j = ( join ‘ 𝐾 )
4 cdlemef46g.m = ( meet ‘ 𝐾 )
5 cdlemef46g.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef46g.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef46g.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef46g.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemefs46g.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
10 cdlemef46g.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
11 cdlemef46.v 𝑉 = ( ( 𝑄 𝑃 ) 𝑊 )
12 cdlemef46.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
13 cdlemefs46.o 𝑂 = ( ( 𝑄 𝑃 ) ( 𝑁 ( ( 𝑢 𝑣 ) 𝑊 ) ) )
14 cdlemef46.g 𝐺 = ( 𝑎𝐵 ↦ if ( ( 𝑄𝑃 ∧ ¬ 𝑎 𝑊 ) , ( 𝑐𝐵𝑢𝐴 ( ( ¬ 𝑢 𝑊 ∧ ( 𝑢 ( 𝑎 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ( 𝑄 𝑃 ) , ( 𝑏𝐵𝑣𝐴 ( ( ¬ 𝑣 𝑊 ∧ ¬ 𝑣 ( 𝑄 𝑃 ) ) → 𝑏 = 𝑂 ) ) , 𝑢 / 𝑣 𝑁 ) ( 𝑎 𝑊 ) ) ) ) , 𝑎 ) )
15 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
16 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → 𝑃𝑄 )
17 simp2rl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → 𝑋𝐵 )
18 vex 𝑠 ∈ V
19 eqid ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
20 8 19 cdleme31sc ( 𝑠 ∈ V → 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) ) )
21 18 20 ax-mp 𝑠 / 𝑡 𝐷 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
22 eqid ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
23 eqid if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 )
24 eqid ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) )
25 1 2 3 4 5 6 7 21 8 9 22 23 24 10 cdleme32fvcl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑋𝐵 ) → ( 𝐹𝑋 ) ∈ 𝐵 )
26 15 17 25 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐹𝑋 ) ∈ 𝐵 )
27 1 2 3 4 5 6 7 8 9 10 cdleme48bw ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ¬ ( 𝐹𝑋 ) 𝑊 )
28 26 27 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝐹𝑋 ) 𝑊 ) )
29 simp3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
30 1 2 3 4 5 6 7 8 9 10 cdleme46fvaw ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) → ( ( 𝐹𝑆 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑆 ) 𝑊 ) )
31 15 29 30 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹𝑆 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑆 ) 𝑊 ) )
32 1 2 3 4 5 6 7 8 9 10 cdleme48b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹𝑋 ) 𝑊 ) = ( 𝑋 𝑊 ) )
33 32 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹𝑆 ) ( ( 𝐹𝑋 ) 𝑊 ) ) = ( ( 𝐹𝑆 ) ( 𝑋 𝑊 ) ) )
34 1 2 3 4 5 6 7 8 9 10 cdleme48fv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐹𝑋 ) = ( ( 𝐹𝑆 ) ( 𝑋 𝑊 ) ) )
35 33 34 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐹𝑆 ) ( ( 𝐹𝑋 ) 𝑊 ) ) = ( 𝐹𝑋 ) )
36 1 2 3 4 5 6 11 12 13 14 cdleme4gfv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( ( 𝐹𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝐹𝑋 ) 𝑊 ) ) ∧ ( ( ( 𝐹𝑆 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑆 ) 𝑊 ) ∧ ( ( 𝐹𝑆 ) ( ( 𝐹𝑋 ) 𝑊 ) ) = ( 𝐹𝑋 ) ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = ( ( 𝐺 ‘ ( 𝐹𝑆 ) ) ( ( 𝐹𝑋 ) 𝑊 ) ) )
37 15 16 28 31 35 36 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = ( ( 𝐺 ‘ ( 𝐹𝑆 ) ) ( ( 𝐹𝑋 ) 𝑊 ) ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdlemeg46gf ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝐺 ‘ ( 𝐹𝑆 ) ) = 𝑆 )
39 15 16 29 38 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ ( 𝐹𝑆 ) ) = 𝑆 )
40 39 32 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐺 ‘ ( 𝐹𝑆 ) ) ( ( 𝐹𝑋 ) 𝑊 ) ) = ( 𝑆 ( 𝑋 𝑊 ) ) )
41 simp3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 )
42 37 40 41 3eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ ( 𝐹𝑋 ) ) = 𝑋 )