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Theorem 2oai1u 804

Description: Orthoarguesian-like OM law.

**Assertion**
Ref	Expression
2oai1u	((a →₁c) ∩ (((a →₁c) ∩ (b →₁c))^⊥ →₂((a^⊥ →₁c) ∩ (b^⊥ →₁c)))) ≤ (b →₁c)

**Proof of Theorem 2oai1u**
Step	Hyp	Ref	Expression
1		1oai1 803	. 2 (((a^⊥ →₁c) →₁c) ∩ (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁(((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))) ≤ ((b^⊥ →₁c) →₁c)
2		u1lem11 762	. . 3 ((a^⊥ →₁c) →₁c) = (a →₁c)
3		u1lem11 762	. . . . . . . 8 ((b^⊥ →₁c) →₁c) = (b →₁c)
4	2, 3	2an 72	. . . . . . 7 (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)) = ((a →₁c) ∩ (b →₁c))
5	4	ax-r1 34	. . . . . 6 ((a →₁c) ∩ (b →₁c)) = (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))
6	5	ud1lem0a 247	. . . . 5 (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁((a →₁c) ∩ (b →₁c))) = (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁(((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))
7	6	ax-r1 34	. . . 4 (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁(((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁((a →₁c) ∩ (b →₁c)))
8		i1i2con2 261	. . . 4 (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁((a →₁c) ∩ (b →₁c))) = (((a →₁c) ∩ (b →₁c))^⊥ →₂((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
9	7, 8	ax-r2 35	. . 3 (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁(((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) = (((a →₁c) ∩ (b →₁c))^⊥ →₂((a^⊥ →₁c) ∩ (b^⊥ →₁c)))
10	2, 9	2an 72	. 2 (((a^⊥ →₁c) →₁c) ∩ (((a^⊥ →₁c) ∩ (b^⊥ →₁c))^⊥ →₁(((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c)))) = ((a →₁c) ∩ (((a →₁c) ∩ (b →₁c))^⊥ →₂((a^⊥ →₁c) ∩ (b^⊥ →₁c))))
11	1, 10, 3	le3tr2 133	1 ((a →₁c) ∩ (((a →₁c) ∩ (b →₁c))^⊥ →₂((a^⊥ →₁c) ∩ (b^⊥ →₁c)))) ≤ (b →₁c)

Colors of variables: term

Syntax hints: ≤ wle 2 ^⊥ wn 4 ∩ wa 7 →₁wi1 13 →₂wi2 14

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-i1 43 df-i2 44 df-le1 122 df-le2 123 df-c1 124 df-c2 125

metamath.org