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Theorem govar 896
 Description: Lemma for converting n-variable Godowski equations to 2n-variable equations.
Hypotheses
Ref Expression
govar.1 ab
govar.2 bc
Assertion
Ref Expression
govar ((ab) ∩ (a2 c)) ≤ (bc)

Proof of Theorem govar
StepHypRef Expression
1 df-i2 45 . . . 4 (a2 c) = (c ∪ (ac ))
21lan 77 . . 3 ((ab) ∩ (a2 c)) = ((ab) ∩ (c ∪ (ac )))
3 ax-a2 31 . . . . 5 (ab) = (ba)
43ran 78 . . . 4 ((ab) ∩ (c ∪ (ac ))) = ((ba) ∩ (c ∪ (ac )))
5 govar.2 . . . . . . . 8 bc
65lecom 180 . . . . . . 7 b C c
76comcom7 460 . . . . . 6 b C c
8 govar.1 . . . . . . . . . . 11 ab
98lecom 180 . . . . . . . . . 10 a C b
109comcom7 460 . . . . . . . . 9 a C b
1110comcom 453 . . . . . . . 8 b C a
1211comcom2 183 . . . . . . 7 b C a
1312, 6com2an 484 . . . . . 6 b C (ac )
147, 13com2or 483 . . . . 5 b C (c ∪ (ac ))
1514, 11fh2r 474 . . . 4 ((ba) ∩ (c ∪ (ac ))) = ((b ∩ (c ∪ (ac ))) ∪ (a ∩ (c ∪ (ac ))))
164, 15ax-r2 36 . . 3 ((ab) ∩ (c ∪ (ac ))) = ((b ∩ (c ∪ (ac ))) ∪ (a ∩ (c ∪ (ac ))))
17 coman1 185 . . . . . . 7 (ac ) C a
1817comcom7 460 . . . . . 6 (ac ) C a
19 coman2 186 . . . . . . 7 (ac ) C c
2019comcom7 460 . . . . . 6 (ac ) C c
2118, 20fh2c 477 . . . . 5 (a ∩ (c ∪ (ac ))) = ((ac) ∪ (a ∩ (ac )))
22 dff 101 . . . . . . . . 9 0 = (aa )
2322ran 78 . . . . . . . 8 (0 ∩ c ) = ((aa ) ∩ c )
2423ax-r1 35 . . . . . . 7 ((aa ) ∩ c ) = (0 ∩ c )
25 anass 76 . . . . . . 7 ((aa ) ∩ c ) = (a ∩ (ac ))
26 an0r 109 . . . . . . 7 (0 ∩ c ) = 0
2724, 25, 263tr2 64 . . . . . 6 (a ∩ (ac )) = 0
2827lor 70 . . . . 5 ((ac) ∪ (a ∩ (ac ))) = ((ac) ∪ 0)
29 or0 102 . . . . 5 ((ac) ∪ 0) = (ac)
3021, 28, 293tr 65 . . . 4 (a ∩ (c ∪ (ac ))) = (ac)
3130lor 70 . . 3 ((b ∩ (c ∪ (ac ))) ∪ (a ∩ (c ∪ (ac )))) = ((b ∩ (c ∪ (ac ))) ∪ (ac))
322, 16, 313tr 65 . 2 ((ab) ∩ (a2 c)) = ((b ∩ (c ∪ (ac ))) ∪ (ac))
33 lea 160 . . 3 (b ∩ (c ∪ (ac ))) ≤ b
34 lear 161 . . 3 (ac) ≤ c
3533, 34le2or 168 . 2 ((b ∩ (c ∪ (ac ))) ∪ (ac)) ≤ (bc)
3632, 35bltr 138 1 ((ab) ∩ (a2 c)) ≤ (bc)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  gon2n  898
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