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Theorem eengtrkg 28752
Description: The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Assertion
Ref Expression
eengtrkg (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ TarskiG)

Proof of Theorem eengtrkg
Dummy variables π‘Ž 𝑏 𝑐 𝑓 𝑖 𝑝 𝑠 𝑑 𝑒 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6900 . . . . . 6 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ V)
2 simpl 482 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
3 simprl 768 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
4 eengbas 28747 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
54adantr 480 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
63, 5eleqtrrd 2830 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
7 simprr 770 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
87, 5eleqtrrd 2830 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
9 axcgrrflx 28680 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ ⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘¦, π‘₯⟩)
102, 6, 8, 9syl3anc 1368 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘¦, π‘₯⟩)
11 eqid 2726 . . . . . . . . 9 (Baseβ€˜(EEGβ€˜π‘)) = (Baseβ€˜(EEGβ€˜π‘))
12 eqid 2726 . . . . . . . . 9 (distβ€˜(EEGβ€˜π‘)) = (distβ€˜(EEGβ€˜π‘))
132, 11, 12, 3, 7, 7, 3ecgrtg 28749 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘¦, π‘₯⟩ ↔ (π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑦(distβ€˜(EEGβ€˜π‘))π‘₯)))
1410, 13mpbid 231 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑦(distβ€˜(EEGβ€˜π‘))π‘₯))
1514ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑦(distβ€˜(EEGβ€˜π‘))π‘₯))
16 simpl 482 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
17 simpr1 1191 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
18 simpr2 1192 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
19 simpr3 1193 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))
2016, 11, 12, 17, 18, 19, 19ecgrtg 28749 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘§, π‘§βŸ© ↔ (π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑧(distβ€˜(EEGβ€˜π‘))𝑧)))
2163adantr3 1168 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
2283adantr3 1168 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
234adantr 480 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
2419, 23eleqtrrd 2830 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
25 axcgrid 28682 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘§, π‘§βŸ© β†’ π‘₯ = 𝑦))
2616, 21, 22, 24, 25syl13anc 1369 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘§, π‘§βŸ© β†’ π‘₯ = 𝑦))
2720, 26sylbird 260 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑧(distβ€˜(EEGβ€˜π‘))𝑧) β†’ π‘₯ = 𝑦))
2827ralrimivvva 3197 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑧(distβ€˜(EEGβ€˜π‘))𝑧) β†’ π‘₯ = 𝑦))
291, 15, 28jca32 515 . . . . 5 (𝑁 ∈ β„• β†’ ((EEGβ€˜π‘) ∈ V ∧ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑦(distβ€˜(EEGβ€˜π‘))π‘₯) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑧(distβ€˜(EEGβ€˜π‘))𝑧) β†’ π‘₯ = 𝑦))))
30 eqid 2726 . . . . . 6 (Itvβ€˜(EEGβ€˜π‘)) = (Itvβ€˜(EEGβ€˜π‘))
3111, 12, 30istrkgc 28213 . . . . 5 ((EEGβ€˜π‘) ∈ TarskiGC ↔ ((EEGβ€˜π‘) ∈ V ∧ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑦(distβ€˜(EEGβ€˜π‘))π‘₯) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (𝑧(distβ€˜(EEGβ€˜π‘))𝑧) β†’ π‘₯ = 𝑦))))
3229, 31sylibr 233 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ TarskiGC)
332, 11, 30, 3, 3, 7ebtwntg 28748 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑦 Btwn ⟨π‘₯, π‘₯⟩ ↔ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯)))
34 axbtwnid 28705 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑦 Btwn ⟨π‘₯, π‘₯⟩ β†’ 𝑦 = π‘₯))
352, 8, 6, 34syl3anc 1368 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑦 Btwn ⟨π‘₯, π‘₯⟩ β†’ 𝑦 = π‘₯))
3633, 35sylbird 260 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯) β†’ 𝑦 = π‘₯))
3736imp 406 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯)) β†’ 𝑦 = π‘₯)
3837equcomd 2014 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯)) β†’ π‘₯ = 𝑦)
3938ex 412 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯) β†’ π‘₯ = 𝑦))
4039ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯) β†’ π‘₯ = 𝑦))
41 simpll 764 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
426adantr 480 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
438adantr 480 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
443adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
457adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
46 simpr1 1191 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))
4741, 44, 45, 46, 24syl13anc 1369 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
48 simpr2 1192 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)))
4941, 4syl 17 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
5048, 49eleqtrrd 2830 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑒 ∈ (π”Όβ€˜π‘))
51 simpr3 1193 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))
5251, 49eleqtrrd 2830 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑣 ∈ (π”Όβ€˜π‘))
53 axpasch 28707 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) ∧ (𝑒 ∈ (π”Όβ€˜π‘) ∧ 𝑣 ∈ (π”Όβ€˜π‘))) β†’ ((𝑒 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑣 Btwn βŸ¨π‘¦, π‘§βŸ©) β†’ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)(π‘Ž Btwn βŸ¨π‘’, π‘¦βŸ© ∧ π‘Ž Btwn βŸ¨π‘£, π‘₯⟩)))
5441, 42, 43, 47, 50, 52, 53syl132anc 1385 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((𝑒 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑣 Btwn βŸ¨π‘¦, π‘§βŸ©) β†’ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)(π‘Ž Btwn βŸ¨π‘’, π‘¦βŸ© ∧ π‘Ž Btwn βŸ¨π‘£, π‘₯⟩)))
5541, 11, 30, 44, 46, 48ebtwntg 28748 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑒 Btwn ⟨π‘₯, π‘§βŸ© ↔ 𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧)))
5641, 11, 30, 45, 46, 51ebtwntg 28748 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑣 Btwn βŸ¨π‘¦, π‘§βŸ© ↔ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)))
5755, 56anbi12d 630 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((𝑒 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑣 Btwn βŸ¨π‘¦, π‘§βŸ©) ↔ (𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧))))
58 simplll 772 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
5948adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)))
6045adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
61 simpr 484 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ π‘Ž ∈ (π”Όβ€˜π‘))
6249adantr 480 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
6361, 62eleqtrd 2829 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))
6458, 11, 30, 59, 60, 63ebtwntg 28748 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ (π‘Ž Btwn βŸ¨π‘’, π‘¦βŸ© ↔ π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦)))
6551adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))
6644adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
6758, 11, 30, 65, 66, 63ebtwntg 28748 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ (π‘Ž Btwn βŸ¨π‘£, π‘₯⟩ ↔ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯)))
6864, 67anbi12d 630 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ ((π‘Ž Btwn βŸ¨π‘’, π‘¦βŸ© ∧ π‘Ž Btwn βŸ¨π‘£, π‘₯⟩) ↔ (π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))))
6949, 68rexeqbidva 3322 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)(π‘Ž Btwn βŸ¨π‘’, π‘¦βŸ© ∧ π‘Ž Btwn βŸ¨π‘£, π‘₯⟩) ↔ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))))
7054, 57, 693imtr3d 293 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)) β†’ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))))
7170ralrimivvva 3197 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))((𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)) β†’ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))))
7271ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))((𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)) β†’ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))))
73 simpl 482 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
74 elpwi 4604 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) β†’ 𝑠 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
7574ad2antrl 725 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑠 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
764adantr 480 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
7775, 76sseqtrrd 4018 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑠 βŠ† (π”Όβ€˜π‘))
78 elpwi 4604 . . . . . . . . . . 11 (𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) β†’ 𝑑 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
7978ad2antll 726 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑑 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
8079, 76sseqtrrd 4018 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑑 βŠ† (π”Όβ€˜π‘))
81 simpll 764 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©) β†’ 𝑁 ∈ β„•)
82 simplrl 774 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©) β†’ 𝑠 βŠ† (π”Όβ€˜π‘))
83 simplrr 775 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©) β†’ 𝑑 βŠ† (π”Όβ€˜π‘))
84 simpr 484 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©) β†’ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©)
85 axcont 28742 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ©)
8681, 82, 83, 84, 85syl13anc 1369 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) ∧ βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ©) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ©)
8786ex 412 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑠 βŠ† (π”Όβ€˜π‘) ∧ 𝑑 βŠ† (π”Όβ€˜π‘))) β†’ (βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ© β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ©))
8873, 77, 80, 87syl12anc 834 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ© β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ©))
89 simplll 772 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑁 ∈ β„•)
90 simplr 766 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘Ž ∈ (π”Όβ€˜π‘))
9176ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
9290, 91eleqtrd 2829 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))
9379ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑑 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
94 simprr 770 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ 𝑑)
9593, 94sseldd 3978 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
9675ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑠 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
97 simprl 768 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ 𝑠)
9896, 97sseldd 3978 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
9989, 11, 30, 92, 95, 98ebtwntg 28748 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ (π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ© ↔ π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦)))
100992ralbidva 3210 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) β†’ (βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ© ↔ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦)))
10176, 100rexeqbidva 3322 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘Ž ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ Btwn βŸ¨π‘Ž, π‘¦βŸ© ↔ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦)))
102 simplll 772 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑁 ∈ β„•)
10375ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑠 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
104 simprl 768 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ 𝑠)
105103, 104sseldd 3978 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
10679ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑑 βŠ† (Baseβ€˜(EEGβ€˜π‘)))
107 simprr 770 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ 𝑑)
108106, 107sseldd 3978 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
109 simplr 766 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑏 ∈ (π”Όβ€˜π‘))
11076ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
111109, 110eleqtrd 2829 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))
112102, 11, 30, 105, 108, 111ebtwntg 28748 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) ∧ (π‘₯ ∈ 𝑠 ∧ 𝑦 ∈ 𝑑)) β†’ (𝑏 Btwn ⟨π‘₯, π‘¦βŸ© ↔ 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))
1131122ralbidva 3210 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑏 ∈ (π”Όβ€˜π‘)) β†’ (βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ© ↔ βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))
11476, 113rexeqbidva 3322 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘ ∈ (π”Όβ€˜π‘)βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 Btwn ⟨π‘₯, π‘¦βŸ© ↔ βˆƒπ‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))
11588, 101, 1143imtr3d 293 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑠 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑑 ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦) β†’ βˆƒπ‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))
116115ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘  ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘‘ ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))(βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦) β†’ βˆƒπ‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))
11740, 72, 1163jca 1125 . . . . 5 (𝑁 ∈ β„• β†’ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))((𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)) β†’ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘‘ ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))(βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦) β†’ βˆƒπ‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦))))
11811, 12, 30istrkgb 28214 . . . . 5 ((EEGβ€˜π‘) ∈ TarskiGB ↔ ((EEGβ€˜π‘) ∈ V ∧ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))((𝑒 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑣 ∈ (𝑦(Itvβ€˜(EEGβ€˜π‘))𝑧)) β†’ βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))(π‘Ž ∈ (𝑒(Itvβ€˜(EEGβ€˜π‘))𝑦) ∧ π‘Ž ∈ (𝑣(Itvβ€˜(EEGβ€˜π‘))π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘‘ ∈ 𝒫 (Baseβ€˜(EEGβ€˜π‘))(βˆƒπ‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑦) β†’ βˆƒπ‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦)))))
1191, 117, 118sylanbrc 582 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ TarskiGB)
12032, 119elind 4189 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ (TarskiGC ∩ TarskiGB))
121 simplll 772 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
1223ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
123121, 4syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
124122, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
1257ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
126125, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
127 simplr1 1212 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))
128127, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
129 simplr2 1213 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)))
130129, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑒 ∈ (π”Όβ€˜π‘))
131 simplr3 1214 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))
132131, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘Ž ∈ (π”Όβ€˜π‘))
133 simpr1 1191 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))
134133, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑏 ∈ (π”Όβ€˜π‘))
135 simpr2 1192 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)))
136135, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑐 ∈ (π”Όβ€˜π‘))
137 simpr3 1193 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))
138137, 123eleqtrrd 2830 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑣 ∈ (π”Όβ€˜π‘))
139 3anass 1092 . . . . . . . . . . . 12 (((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©)) ↔ ((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ ((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))))
140 ax5seg 28704 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑒 ∈ (π”Όβ€˜π‘) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (𝑏 ∈ (π”Όβ€˜π‘) ∧ 𝑐 ∈ (π”Όβ€˜π‘) ∧ 𝑣 ∈ (π”Όβ€˜π‘))) β†’ (((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©)) β†’ βŸ¨π‘§, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))
141139, 140biimtrrid 242 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑒 ∈ (π”Όβ€˜π‘) ∧ π‘Ž ∈ (π”Όβ€˜π‘)) ∧ (𝑏 ∈ (π”Όβ€˜π‘) ∧ 𝑐 ∈ (π”Όβ€˜π‘) ∧ 𝑣 ∈ (π”Όβ€˜π‘))) β†’ (((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ ((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))) β†’ βŸ¨π‘§, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))
142121, 124, 126, 128, 130, 132, 134, 136, 138, 141syl333anc 1399 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ ((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))) β†’ βŸ¨π‘§, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))
143121, 11, 30, 122, 127, 125ebtwntg 28748 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑦 Btwn ⟨π‘₯, π‘§βŸ© ↔ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧)))
144121, 11, 30, 131, 135, 133ebtwntg 28748 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (𝑏 Btwn βŸ¨π‘Ž, π‘βŸ© ↔ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)))
145143, 1443anbi23d 1435 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ↔ (π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐))))
146121, 11, 12, 122, 125, 131, 133ecgrtg 28749 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ↔ (π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))
147121, 11, 12, 125, 127, 133, 135ecgrtg 28749 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ© ↔ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)))
148146, 147anbi12d 630 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ↔ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐))))
149121, 11, 12, 122, 129, 131, 137ecgrtg 28749 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ↔ (π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣)))
150121, 11, 12, 125, 129, 133, 137ecgrtg 28749 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ© ↔ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))
151149, 150anbi12d 630 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ ((⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©) ↔ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣))))
152148, 151anbi12d 630 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©)) ↔ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))))
153145, 152anbi12d 630 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (((π‘₯ β‰  𝑦 ∧ 𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ 𝑏 Btwn βŸ¨π‘Ž, π‘βŸ©) ∧ ((⟨π‘₯, π‘¦βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘, π‘βŸ©) ∧ (⟨π‘₯, π‘’βŸ©CgrβŸ¨π‘Ž, π‘£βŸ© ∧ βŸ¨π‘¦, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ©))) ↔ ((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣))))))
154121, 11, 12, 127, 129, 135, 137ecgrtg 28749 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βŸ¨π‘§, π‘’βŸ©CgrβŸ¨π‘, π‘£βŸ© ↔ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)))
155142, 153, 1543imtr3d 293 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑐 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑣 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)))
156155ralrimivvva 3197 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑒 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))(((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)))
157156ralrimivvva 3197 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))(((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)))
158157ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))(((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)))
159 simpll 764 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑁 ∈ β„•)
1606adantr 480 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
1618adantr 480 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
162 simprl 768 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))
163159, 4syl 17 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
164162, 163eleqtrrd 2830 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ π‘Ž ∈ (π”Όβ€˜π‘))
165 simprr 770 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))
166165, 163eleqtrrd 2830 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ 𝑏 ∈ (π”Όβ€˜π‘))
167 axsegcon 28693 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (π‘Ž ∈ (π”Όβ€˜π‘) ∧ 𝑏 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘Ž, π‘βŸ©))
168159, 160, 161, 164, 166, 167syl122anc 1376 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘Ž, π‘βŸ©))
169 simplll 772 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
1703ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)))
171 simpr 484 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
172163adantr 480 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
173171, 172eleqtrd 2829 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)))
1747ad2antrr 723 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))
175169, 11, 30, 170, 173, 174ebtwntg 28748 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (𝑦 Btwn ⟨π‘₯, π‘§βŸ© ↔ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧)))
176 simplrl 774 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)))
177 simplrr 775 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))
178169, 11, 12, 174, 173, 176, 177ecgrtg 28749 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘Ž, π‘βŸ© ↔ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))
179175, 178anbi12d 630 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ ((𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘Ž, π‘βŸ©) ↔ (𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏))))
180163, 179rexeqbidva 3322 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ (βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨π‘₯, π‘§βŸ© ∧ βŸ¨π‘¦, π‘§βŸ©CgrβŸ¨π‘Ž, π‘βŸ©) ↔ βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏))))
181168, 180mpbid 231 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) ∧ (π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑏 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))
182181ralrimivva 3194 . . . . . . 7 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)) ∧ 𝑦 ∈ (Baseβ€˜(EEGβ€˜π‘)))) β†’ βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))
183182ralrimivva 3194 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))
1841, 158, 183jca32 515 . . . . 5 (𝑁 ∈ β„• β†’ ((EEGβ€˜π‘) ∈ V ∧ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))(((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))))
18511, 12, 30istrkgcb 28215 . . . . 5 ((EEGβ€˜π‘) ∈ TarskiGCB ↔ ((EEGβ€˜π‘) ∈ V ∧ (βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘’ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘£ ∈ (Baseβ€˜(EEGβ€˜π‘))(((π‘₯ β‰  𝑦 ∧ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ 𝑏 ∈ (π‘Ž(Itvβ€˜(EEGβ€˜π‘))𝑐)) ∧ (((π‘₯(distβ€˜(EEGβ€˜π‘))𝑦) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑐)) ∧ ((π‘₯(distβ€˜(EEGβ€˜π‘))𝑒) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑣) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑏(distβ€˜(EEGβ€˜π‘))𝑣)))) β†’ (𝑧(distβ€˜(EEGβ€˜π‘))𝑒) = (𝑐(distβ€˜(EEGβ€˜π‘))𝑣)) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘¦ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘Ž ∈ (Baseβ€˜(EEGβ€˜π‘))βˆ€π‘ ∈ (Baseβ€˜(EEGβ€˜π‘))βˆƒπ‘§ ∈ (Baseβ€˜(EEGβ€˜π‘))(𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧) ∧ (𝑦(distβ€˜(EEGβ€˜π‘))𝑧) = (π‘Ž(distβ€˜(EEGβ€˜π‘))𝑏)))))
186184, 185sylibr 233 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ TarskiGCB)
18711, 30elntg 28750 . . . . 5 (𝑁 ∈ β„• β†’ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)), 𝑦 ∈ ((Baseβ€˜(EEGβ€˜π‘)) βˆ– {π‘₯}) ↦ {𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∣ (𝑧 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦) ∨ π‘₯ ∈ (𝑧(Itvβ€˜(EEGβ€˜π‘))𝑦) ∨ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧))}))
18811, 12, 30istrkgl 28217 . . . . 5 ((EEGβ€˜π‘) ∈ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})} ↔ ((EEGβ€˜π‘) ∈ V ∧ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ (Baseβ€˜(EEGβ€˜π‘)), 𝑦 ∈ ((Baseβ€˜(EEGβ€˜π‘)) βˆ– {π‘₯}) ↦ {𝑧 ∈ (Baseβ€˜(EEGβ€˜π‘)) ∣ (𝑧 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑦) ∨ π‘₯ ∈ (𝑧(Itvβ€˜(EEGβ€˜π‘))𝑦) ∨ 𝑦 ∈ (π‘₯(Itvβ€˜(EEGβ€˜π‘))𝑧))})))
1891, 187, 188sylanbrc 582 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})})
190186, 189elind 4189 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})}))
191120, 190elind 4189 . 2 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})})))
192 df-trkg 28212 . 2 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})}))
193191, 192eleqtrrdi 2838 1 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ TarskiG)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426  Vcvv 3468  [wsbc 3772   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  {csn 4623  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  β„•cn 12216  Basecbs 17153  distcds 17215  TarskiGcstrkg 28186  TarskiGCcstrkgc 28187  TarskiGBcstrkgb 28188  TarskiGCBcstrkgcb 28189  Itvcitv 28192  LineGclng 28193  π”Όcee 28654   Btwn cbtwn 28655  Cgrccgr 28656  EEGceeng 28743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-rp 12981  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-ds 17228  df-itv 28194  df-lng 28195  df-trkgc 28207  df-trkgb 28208  df-trkgcb 28209  df-trkg 28212  df-ee 28657  df-btwn 28658  df-cgr 28659  df-eeng 28744
This theorem is referenced by: (None)
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