Metamath Proof Explorer


Theorem goeqi

Description: Godowski's equation, shown here as a variant equivalent to Equation SF of Godowski p. 730. (Contributed by NM, 10-Nov-2002) (New usage is discouraged.)

Ref Expression
Hypotheses goeq.1 𝐴C
goeq.2 𝐵C
goeq.3 𝐶C
goeq.4 𝐹 = ( ( ⊥ ‘ 𝐴 ) ∨ ( 𝐴𝐵 ) )
goeq.5 𝐺 = ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐶 ) )
goeq.6 𝐻 = ( ( ⊥ ‘ 𝐶 ) ∨ ( 𝐶𝐴 ) )
goeq.7 𝐷 = ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐴 ) )
Assertion goeqi ( ( 𝐹𝐺 ) ∩ 𝐻 ) ⊆ 𝐷

Proof

Step Hyp Ref Expression
1 goeq.1 𝐴C
2 goeq.2 𝐵C
3 goeq.3 𝐶C
4 goeq.4 𝐹 = ( ( ⊥ ‘ 𝐴 ) ∨ ( 𝐴𝐵 ) )
5 goeq.5 𝐺 = ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐶 ) )
6 goeq.6 𝐻 = ( ( ⊥ ‘ 𝐶 ) ∨ ( 𝐶𝐴 ) )
7 goeq.7 𝐷 = ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐴 ) )
8 1 choccli ( ⊥ ‘ 𝐴 ) ∈ C
9 1 2 chincli ( 𝐴𝐵 ) ∈ C
10 8 9 chjcli ( ( ⊥ ‘ 𝐴 ) ∨ ( 𝐴𝐵 ) ) ∈ C
11 4 10 eqeltri 𝐹C
12 2 choccli ( ⊥ ‘ 𝐵 ) ∈ C
13 2 3 chincli ( 𝐵𝐶 ) ∈ C
14 12 13 chjcli ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐶 ) ) ∈ C
15 5 14 eqeltri 𝐺C
16 11 15 chincli ( 𝐹𝐺 ) ∈ C
17 3 choccli ( ⊥ ‘ 𝐶 ) ∈ C
18 3 1 chincli ( 𝐶𝐴 ) ∈ C
19 17 18 chjcli ( ( ⊥ ‘ 𝐶 ) ∨ ( 𝐶𝐴 ) ) ∈ C
20 6 19 eqeltri 𝐻C
21 16 20 chincli ( ( 𝐹𝐺 ) ∩ 𝐻 ) ∈ C
22 2 1 chincli ( 𝐵𝐴 ) ∈ C
23 12 22 chjcli ( ( ⊥ ‘ 𝐵 ) ∨ ( 𝐵𝐴 ) ) ∈ C
24 7 23 eqeltri 𝐷C
25 21 24 stri ( ∀ 𝑓 ∈ States ( ( 𝑓 ‘ ( ( 𝐹𝐺 ) ∩ 𝐻 ) ) = 1 → ( 𝑓𝐷 ) = 1 ) → ( ( 𝐹𝐺 ) ∩ 𝐻 ) ⊆ 𝐷 )
26 eqid ( ( ⊥ ‘ 𝐶 ) ∨ ( 𝐶𝐵 ) ) = ( ( ⊥ ‘ 𝐶 ) ∨ ( 𝐶𝐵 ) )
27 eqid ( ( ⊥ ‘ 𝐴 ) ∨ ( 𝐴𝐶 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ ( 𝐴𝐶 ) )
28 1 2 3 4 5 6 7 26 27 golem2 ( 𝑓 ∈ States → ( ( 𝑓 ‘ ( ( 𝐹𝐺 ) ∩ 𝐻 ) ) = 1 → ( 𝑓𝐷 ) = 1 ) )
29 25 28 mprg ( ( 𝐹𝐺 ) ∩ 𝐻 ) ⊆ 𝐷