Theorem govar 876

Description: Lemma for converting n-variable Godowski equations to 2n-variable equations

Hypotheses

Ref	Expression
govar.1	a =< b_|_
govar.2	b =< c_|_

Assertion

Ref	Expression
govar	((a v b) ^ (a ->2 c)) =< (b v c)

Proof of Theorem govar

Step	Hyp	Ref	Expression
1		df-i2 44	. . . 4 (a ->2 c) = (c v (a_|_ ^ c_|_))
2	1	lan 70	. . 3 ((a v b) ^ (a ->2 c)) = ((a v b) ^ (c v (a_|_ ^ c_|_)))
3		ax-a2 30	. . . . 5 (a v b) = (b v a)
4	3	ran 71	. . . 4 ((a v b) ^ (c v (a_|_ ^ c_|_))) = ((b v a) ^ (c v (a_|_ ^ c_|_)))
5		govar.2	. . . . . . . 8 b =< c_|_
6	5	lecom 172	. . . . . . 7 b C c_|_
7	6	comcom7 442	. . . . . 6 b C c
8		govar.1	. . . . . . . . . . 11 a =< b_|_
9	8	lecom 172	. . . . . . . . . 10 a C b_|_
10	9	comcom7 442	. . . . . . . . 9 a C b
11	10	comcom 435	. . . . . . . 8 b C a
12	11	comcom2 175	. . . . . . 7 b C a_|_
13	12, 6	com2an 466	. . . . . 6 b C (a_|_ ^ c_|_)
14	7, 13	com2or 465	. . . . 5 b C (c v (a_|_ ^ c_|_))
15	14, 11	fh2r 456	. . . 4 ((b v a) ^ (c v (a_|_ ^ c_|_))) = ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ (c v (a_|_ ^ c_|_))))
16	4, 15	ax-r2 35	. . 3 ((a v b) ^ (c v (a_|_ ^ c_|_))) = ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ (c v (a_|_ ^ c_|_))))
17		coman1 177	. . . . . . 7 (a_|_ ^ c_|_) C a_|_
18	17	comcom7 442	. . . . . 6 (a_|_ ^ c_|_) C a
19		coman2 178	. . . . . . 7 (a_|_ ^ c_|_) C c_|_
20	19	comcom7 442	. . . . . 6 (a_|_ ^ c_|_) C c
21	18, 20	fh2c 459	. . . . 5 (a ^ (c v (a_|_ ^ c_|_))) = ((a ^ c) v (a ^ (a_|_ ^ c_|_)))
22		dff 93	. . . . . . . . 9 0 = (a ^ a_|_)
23	22	ran 71	. . . . . . . 8 (0 ^ c_|_) = ((a ^ a_|_) ^ c_|_)
24	23	ax-r1 34	. . . . . . 7 ((a ^ a_|_) ^ c_|_) = (0 ^ c_|_)
25		anass 69	. . . . . . 7 ((a ^ a_|_) ^ c_|_) = (a ^ (a_|_ ^ c_|_))
26		an0r 101	. . . . . . 7 (0 ^ c_|_) = 0
27	24, 25, 26	3tr2 61	. . . . . 6 (a ^ (a_|_ ^ c_|_)) = 0
28	27	lor 66	. . . . 5 ((a ^ c) v (a ^ (a_|_ ^ c_|_))) = ((a ^ c) v 0)
29		or0 94	. . . . 5 ((a ^ c) v 0) = (a ^ c)
30	21, 28, 29	3tr 62	. . . 4 (a ^ (c v (a_|_ ^ c_|_))) = (a ^ c)
31	30	lor 66	. . 3 ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ (c v (a_|_ ^ c_|_)))) = ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ c))
32	2, 16, 31	3tr 62	. 2 ((a v b) ^ (a ->2 c)) = ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ c))
33		lea 152	. . 3 (b ^ (c v (a_|_ ^ c_|_))) =< b
34		lear 153	. . 3 (a ^ c) =< c
35	33, 34	le2or 160	. 2 ((b ^ (c v (a_|_ ^ c_|_))) v (a ^ c)) =< (b v c)
36	32, 35	bltr 130	1 ((a v b) ^ (a ->2 c)) =< (b v c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->2 wi2 14

This theorem is referenced by:  gon2n 878

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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