Step |
Hyp |
Ref |
Expression |
1 |
|
axtrkg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
axtrkg.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
axtrkg.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
axtrkg.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
axtgcont.1 |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑃 ) |
6 |
|
axtgcont.2 |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑃 ) |
7 |
|
df-trkg |
⊢ TarskiG = ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓 ∣ [ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥 ∈ 𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧 ∈ 𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ) |
8 |
|
inss1 |
⊢ ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓 ∣ [ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥 ∈ 𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧 ∈ 𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ) ⊆ ( TarskiGC ∩ TarskiGB ) |
9 |
|
inss2 |
⊢ ( TarskiGC ∩ TarskiGB ) ⊆ TarskiGB |
10 |
8 9
|
sstri |
⊢ ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓 ∣ [ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥 ∈ 𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧 ∈ 𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ) ⊆ TarskiGB |
11 |
7 10
|
eqsstri |
⊢ TarskiG ⊆ TarskiGB |
12 |
11 4
|
sselid |
⊢ ( 𝜑 → 𝐺 ∈ TarskiGB ) |
13 |
1 2 3
|
istrkgb |
⊢ ( 𝐺 ∈ TarskiGB ↔ ( 𝐺 ∈ V ∧ ( ∀ 𝑥 ∈ 𝑃 ∀ 𝑦 ∈ 𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑥 ) → 𝑥 = 𝑦 ) ∧ ∀ 𝑥 ∈ 𝑃 ∀ 𝑦 ∈ 𝑃 ∀ 𝑧 ∈ 𝑃 ∀ 𝑢 ∈ 𝑃 ∀ 𝑣 ∈ 𝑃 ( ( 𝑢 ∈ ( 𝑥 𝐼 𝑧 ) ∧ 𝑣 ∈ ( 𝑦 𝐼 𝑧 ) ) → ∃ 𝑎 ∈ 𝑃 ( 𝑎 ∈ ( 𝑢 𝐼 𝑦 ) ∧ 𝑎 ∈ ( 𝑣 𝐼 𝑥 ) ) ) ∧ ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) ) |
14 |
13
|
simprbi |
⊢ ( 𝐺 ∈ TarskiGB → ( ∀ 𝑥 ∈ 𝑃 ∀ 𝑦 ∈ 𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑥 ) → 𝑥 = 𝑦 ) ∧ ∀ 𝑥 ∈ 𝑃 ∀ 𝑦 ∈ 𝑃 ∀ 𝑧 ∈ 𝑃 ∀ 𝑢 ∈ 𝑃 ∀ 𝑣 ∈ 𝑃 ( ( 𝑢 ∈ ( 𝑥 𝐼 𝑧 ) ∧ 𝑣 ∈ ( 𝑦 𝐼 𝑧 ) ) → ∃ 𝑎 ∈ 𝑃 ( 𝑎 ∈ ( 𝑢 𝐼 𝑦 ) ∧ 𝑎 ∈ ( 𝑣 𝐼 𝑥 ) ) ) ∧ ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) |
15 |
14
|
simp3d |
⊢ ( 𝐺 ∈ TarskiGB → ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
16 |
12 15
|
syl |
⊢ ( 𝜑 → ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
17 |
1
|
fvexi |
⊢ 𝑃 ∈ V |
18 |
17
|
ssex |
⊢ ( 𝑆 ⊆ 𝑃 → 𝑆 ∈ V ) |
19 |
|
elpwg |
⊢ ( 𝑆 ∈ V → ( 𝑆 ∈ 𝒫 𝑃 ↔ 𝑆 ⊆ 𝑃 ) ) |
20 |
5 18 19
|
3syl |
⊢ ( 𝜑 → ( 𝑆 ∈ 𝒫 𝑃 ↔ 𝑆 ⊆ 𝑃 ) ) |
21 |
5 20
|
mpbird |
⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝑃 ) |
22 |
17
|
ssex |
⊢ ( 𝑇 ⊆ 𝑃 → 𝑇 ∈ V ) |
23 |
|
elpwg |
⊢ ( 𝑇 ∈ V → ( 𝑇 ∈ 𝒫 𝑃 ↔ 𝑇 ⊆ 𝑃 ) ) |
24 |
6 22 23
|
3syl |
⊢ ( 𝜑 → ( 𝑇 ∈ 𝒫 𝑃 ↔ 𝑇 ⊆ 𝑃 ) ) |
25 |
6 24
|
mpbird |
⊢ ( 𝜑 → 𝑇 ∈ 𝒫 𝑃 ) |
26 |
|
raleq |
⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ) ) |
27 |
26
|
rexbidv |
⊢ ( 𝑠 = 𝑆 → ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ↔ ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ) ) |
28 |
|
raleq |
⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
29 |
28
|
rexbidv |
⊢ ( 𝑠 = 𝑆 → ( ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ↔ ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
30 |
27 29
|
imbi12d |
⊢ ( 𝑠 = 𝑆 → ( ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ↔ ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) |
31 |
|
raleq |
⊢ ( 𝑡 = 𝑇 → ( ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ) ) |
32 |
31
|
rexralbidv |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ↔ ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) ) ) |
33 |
|
raleq |
⊢ ( 𝑡 = 𝑇 → ( ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
34 |
33
|
rexralbidv |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ↔ ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |
35 |
32 34
|
imbi12d |
⊢ ( 𝑡 = 𝑇 → ( ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ↔ ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) |
36 |
30 35
|
rspc2v |
⊢ ( ( 𝑆 ∈ 𝒫 𝑃 ∧ 𝑇 ∈ 𝒫 𝑃 ) → ( ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) → ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) |
37 |
21 25 36
|
syl2anc |
⊢ ( 𝜑 → ( ∀ 𝑠 ∈ 𝒫 𝑃 ∀ 𝑡 ∈ 𝒫 𝑃 ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑡 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) → ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) ) |
38 |
16 37
|
mpd |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑥 ∈ ( 𝑎 𝐼 𝑦 ) → ∃ 𝑏 ∈ 𝑃 ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 𝑏 ∈ ( 𝑥 𝐼 𝑦 ) ) ) |