Metamath Proof Explorer


Theorem chocunii

Description: Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of Beran p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999) (Proof shortened by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

Ref Expression
Hypothesis chocuni.1 H C
Assertion chocunii A H B H C H D H R = A + B R = C + D A = C B = D

Proof

Step Hyp Ref Expression
1 chocuni.1 H C
2 1 chshii H S
3 2 a1i A H B H C H D H R = A + B R = C + D H S
4 shocsh H S H S
5 2 4 mp1i A H B H C H D H R = A + B R = C + D H S
6 ocin H S H H = 0
7 2 6 mp1i A H B H C H D H R = A + B R = C + D H H = 0
8 simplll A H B H C H D H R = A + B R = C + D A H
9 simpllr A H B H C H D H R = A + B R = C + D B H
10 simplrl A H B H C H D H R = A + B R = C + D C H
11 simplrr A H B H C H D H R = A + B R = C + D D H
12 eqtr2 R = A + B R = C + D A + B = C + D
13 12 adantl A H B H C H D H R = A + B R = C + D A + B = C + D
14 3 5 7 8 9 10 11 13 shuni A H B H C H D H R = A + B R = C + D A = C B = D
15 14 ex A H B H C H D H R = A + B R = C + D A = C B = D