Step |
Hyp |
Ref |
Expression |
1 |
|
phtpcco2.f |
⊢ ( 𝜑 → 𝐹 ( ≃ph ‘ 𝐽 ) 𝐺 ) |
2 |
|
phtpcco2.p |
⊢ ( 𝜑 → 𝑃 ∈ ( 𝐽 Cn 𝐾 ) ) |
3 |
|
isphtpc |
⊢ ( 𝐹 ( ≃ph ‘ 𝐽 ) 𝐺 ↔ ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ 𝐺 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ≠ ∅ ) ) |
4 |
1 3
|
sylib |
⊢ ( 𝜑 → ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ 𝐺 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ≠ ∅ ) ) |
5 |
4
|
simp1d |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
6 |
|
cnco |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ 𝑃 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝑃 ∘ 𝐹 ) ∈ ( II Cn 𝐾 ) ) |
7 |
5 2 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ∘ 𝐹 ) ∈ ( II Cn 𝐾 ) ) |
8 |
4
|
simp2d |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
9 |
|
cnco |
⊢ ( ( 𝐺 ∈ ( II Cn 𝐽 ) ∧ 𝑃 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝑃 ∘ 𝐺 ) ∈ ( II Cn 𝐾 ) ) |
10 |
8 2 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ∘ 𝐺 ) ∈ ( II Cn 𝐾 ) ) |
11 |
4
|
simp3d |
⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ≠ ∅ ) |
12 |
|
n0 |
⊢ ( ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) |
13 |
11 12
|
sylib |
⊢ ( 𝜑 → ∃ 𝑓 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) |
14 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → 𝐹 ∈ ( II Cn 𝐽 ) ) |
15 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → 𝐺 ∈ ( II Cn 𝐽 ) ) |
16 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → 𝑃 ∈ ( 𝐽 Cn 𝐾 ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) |
18 |
14 15 16 17
|
phtpyco2 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → ( 𝑃 ∘ 𝑓 ) ∈ ( ( 𝑃 ∘ 𝐹 ) ( PHtpy ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ) ) |
19 |
18
|
ne0d |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ) → ( ( 𝑃 ∘ 𝐹 ) ( PHtpy ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ) ≠ ∅ ) |
20 |
13 19
|
exlimddv |
⊢ ( 𝜑 → ( ( 𝑃 ∘ 𝐹 ) ( PHtpy ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ) ≠ ∅ ) |
21 |
|
isphtpc |
⊢ ( ( 𝑃 ∘ 𝐹 ) ( ≃ph ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ↔ ( ( 𝑃 ∘ 𝐹 ) ∈ ( II Cn 𝐾 ) ∧ ( 𝑃 ∘ 𝐺 ) ∈ ( II Cn 𝐾 ) ∧ ( ( 𝑃 ∘ 𝐹 ) ( PHtpy ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ) ≠ ∅ ) ) |
22 |
7 10 20 21
|
syl3anbrc |
⊢ ( 𝜑 → ( 𝑃 ∘ 𝐹 ) ( ≃ph ‘ 𝐾 ) ( 𝑃 ∘ 𝐺 ) ) |