Metamath Proof Explorer


Theorem sgrp2nmndlem3

Description: Lemma 3 for sgrp2nmnd . (Contributed by AV, 29-Jan-2020)

Ref Expression
Hypotheses mgm2nsgrp.s 𝑆 = { 𝐴 , 𝐵 }
mgm2nsgrp.b ( Base ‘ 𝑀 ) = 𝑆
sgrp2nmnd.o ( +g𝑀 ) = ( 𝑥𝑆 , 𝑦𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) )
sgrp2nmnd.p = ( +g𝑀 )
Assertion sgrp2nmndlem3 ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → ( 𝐵 𝐶 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 mgm2nsgrp.s 𝑆 = { 𝐴 , 𝐵 }
2 mgm2nsgrp.b ( Base ‘ 𝑀 ) = 𝑆
3 sgrp2nmnd.o ( +g𝑀 ) = ( 𝑥𝑆 , 𝑦𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) )
4 sgrp2nmnd.p = ( +g𝑀 )
5 4 3 eqtri = ( 𝑥𝑆 , 𝑦𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) )
6 5 a1i ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → = ( 𝑥𝑆 , 𝑦𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) ) )
7 df-ne ( 𝐴𝐵 ↔ ¬ 𝐴 = 𝐵 )
8 eqeq2 ( 𝑥 = 𝐵 → ( 𝐴 = 𝑥𝐴 = 𝐵 ) )
9 8 adantr ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ( 𝐴 = 𝑥𝐴 = 𝐵 ) )
10 eqcom ( 𝐴 = 𝑥𝑥 = 𝐴 )
11 9 10 bitr3di ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ( 𝐴 = 𝐵𝑥 = 𝐴 ) )
12 11 notbid ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ( ¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴 ) )
13 12 biimpcd ( ¬ 𝐴 = 𝐵 → ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) )
14 7 13 sylbi ( 𝐴𝐵 → ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) )
15 14 3ad2ant3 ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → ( ( 𝑥 = 𝐵𝑦 = 𝐶 ) → ¬ 𝑥 = 𝐴 ) )
16 15 imp ( ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) ∧ ( 𝑥 = 𝐵𝑦 = 𝐶 ) ) → ¬ 𝑥 = 𝐴 )
17 16 iffalsed ( ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) ∧ ( 𝑥 = 𝐵𝑦 = 𝐶 ) ) → if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) = 𝐵 )
18 simp2 ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → 𝐵𝑆 )
19 simp1 ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → 𝐶𝑆 )
20 6 17 18 19 18 ovmpod ( ( 𝐶𝑆𝐵𝑆𝐴𝐵 ) → ( 𝐵 𝐶 ) = 𝐵 )