Step |
Hyp |
Ref |
Expression |
1 |
|
mndtccat.c |
⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) ) |
2 |
|
mndtccat.m |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
3 |
|
oppgoppchom.d |
⊢ ( 𝜑 → 𝐷 = ( MndToCat ‘ ( oppg ‘ 𝑀 ) ) ) |
4 |
|
oppgoppchom.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
5 |
|
oppgoppchom.x |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐷 ) ) |
6 |
|
oppgoppchom.y |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝑂 ) ) |
7 |
|
oppgoppcco.o |
⊢ ( 𝜑 → · = ( comp ‘ 𝐷 ) ) |
8 |
|
oppgoppcco.x |
⊢ ( 𝜑 → ∙ = ( comp ‘ 𝑂 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
10 |
4 9
|
oppcbas |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝑂 ) |
11 |
10
|
eqcomi |
⊢ ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 ) |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑂 ) = ( Base ‘ 𝐶 ) ) |
13 |
|
eqidd |
⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) ) |
14 |
1 2 12 6 6 6 13
|
mndtcco |
⊢ ( 𝜑 → ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑌 ) = ( +g ‘ 𝑀 ) ) |
15 |
14
|
tposeqd |
⊢ ( 𝜑 → tpos ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑌 ) = tpos ( +g ‘ 𝑀 ) ) |
16 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
17 |
11 16 4 6 6 6
|
oppccofval |
⊢ ( 𝜑 → ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝑂 ) 𝑌 ) = tpos ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑌 ) ) |
18 |
|
eqid |
⊢ ( oppg ‘ 𝑀 ) = ( oppg ‘ 𝑀 ) |
19 |
18
|
oppgmnd |
⊢ ( 𝑀 ∈ Mnd → ( oppg ‘ 𝑀 ) ∈ Mnd ) |
20 |
2 19
|
syl |
⊢ ( 𝜑 → ( oppg ‘ 𝑀 ) ∈ Mnd ) |
21 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) ) |
22 |
3 20 21 5 5 5 7
|
mndtcco |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑋 〉 · 𝑋 ) = ( +g ‘ ( oppg ‘ 𝑀 ) ) ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
24 |
|
eqid |
⊢ ( +g ‘ ( oppg ‘ 𝑀 ) ) = ( +g ‘ ( oppg ‘ 𝑀 ) ) |
25 |
23 18 24
|
oppgplusfval |
⊢ ( +g ‘ ( oppg ‘ 𝑀 ) ) = tpos ( +g ‘ 𝑀 ) |
26 |
22 25
|
eqtrdi |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑋 〉 · 𝑋 ) = tpos ( +g ‘ 𝑀 ) ) |
27 |
15 17 26
|
3eqtr4rd |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑋 〉 · 𝑋 ) = ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝑂 ) 𝑌 ) ) |
28 |
8
|
oveqd |
⊢ ( 𝜑 → ( 〈 𝑌 , 𝑌 〉 ∙ 𝑌 ) = ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝑂 ) 𝑌 ) ) |
29 |
27 28
|
eqtr4d |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑋 〉 · 𝑋 ) = ( 〈 𝑌 , 𝑌 〉 ∙ 𝑌 ) ) |