Step |
Hyp |
Ref |
Expression |
1 |
|
legval.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
legval.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
legval.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
legval.l |
⊢ ≤ = ( ≤G ‘ 𝐺 ) |
5 |
|
legval.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
6 |
|
legso.a |
⊢ 𝐸 = ( − “ ( 𝑃 × 𝑃 ) ) |
7 |
|
legso.f |
⊢ ( 𝜑 → Fun − ) |
8 |
|
ltgseg.p |
⊢ ( 𝜑 → 𝐴 ∈ 𝐸 ) |
9 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑎 = 〈 𝑥 , 𝑦 〉 ) → ( − ‘ 𝑎 ) = 𝐴 ) |
10 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑎 = 〈 𝑥 , 𝑦 〉 ) → 𝑎 = 〈 𝑥 , 𝑦 〉 ) |
11 |
10
|
fveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑎 = 〈 𝑥 , 𝑦 〉 ) → ( − ‘ 𝑎 ) = ( − ‘ 〈 𝑥 , 𝑦 〉 ) ) |
12 |
9 11
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑎 = 〈 𝑥 , 𝑦 〉 ) → 𝐴 = ( − ‘ 〈 𝑥 , 𝑦 〉 ) ) |
13 |
|
df-ov |
⊢ ( 𝑥 − 𝑦 ) = ( − ‘ 〈 𝑥 , 𝑦 〉 ) |
14 |
12 13
|
eqtr4di |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ 𝑎 = 〈 𝑥 , 𝑦 〉 ) → 𝐴 = ( 𝑥 − 𝑦 ) ) |
15 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) → 𝑎 ∈ ( 𝑃 × 𝑃 ) ) |
16 |
|
elxp2 |
⊢ ( 𝑎 ∈ ( 𝑃 × 𝑃 ) ↔ ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 𝑎 = 〈 𝑥 , 𝑦 〉 ) |
17 |
15 16
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) → ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 𝑎 = 〈 𝑥 , 𝑦 〉 ) |
18 |
14 17
|
reximddv2 |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝑃 × 𝑃 ) ) ∧ ( − ‘ 𝑎 ) = 𝐴 ) → ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 𝐴 = ( 𝑥 − 𝑦 ) ) |
19 |
8 6
|
eleqtrdi |
⊢ ( 𝜑 → 𝐴 ∈ ( − “ ( 𝑃 × 𝑃 ) ) ) |
20 |
|
fvelima |
⊢ ( ( Fun − ∧ 𝐴 ∈ ( − “ ( 𝑃 × 𝑃 ) ) ) → ∃ 𝑎 ∈ ( 𝑃 × 𝑃 ) ( − ‘ 𝑎 ) = 𝐴 ) |
21 |
7 19 20
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ( 𝑃 × 𝑃 ) ( − ‘ 𝑎 ) = 𝐴 ) |
22 |
18 21
|
r19.29a |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 𝐴 = ( 𝑥 − 𝑦 ) ) |