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Theorem elntg 28782
Description: The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)
Hypotheses
Ref Expression
elntg.1 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
elntg.2 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
Assertion
Ref Expression
elntg (𝑁 ∈ β„• β†’ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝑁   𝑧,𝑃
Allowed substitution hints:   𝑃(π‘₯,𝑦)   𝐼(π‘₯,𝑦,𝑧)

Proof of Theorem elntg
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 lngid 28231 . . 3 LineG = Slot (LineGβ€˜ndx)
2 fvex 6904 . . . 4 (EEGβ€˜π‘) ∈ V
32a1i 11 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ V)
4 eengstr 28778 . . . . 5 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
5 structn0fun 17111 . . . . 5 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
64, 5syl 17 . . . 4 (𝑁 ∈ β„• β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
7 structcnvcnv 17113 . . . . . 6 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
84, 7syl 17 . . . . 5 (𝑁 ∈ β„• β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
98funeqd 6569 . . . 4 (𝑁 ∈ β„• β†’ (Fun β—‘β—‘(EEGβ€˜π‘) ↔ Fun ((EEGβ€˜π‘) βˆ– {βˆ…})))
106, 9mpbird 257 . . 3 (𝑁 ∈ β„• β†’ Fun β—‘β—‘(EEGβ€˜π‘))
11 opex 5460 . . . . . 6 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ V
1211prid2 4763 . . . . 5 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}
13 elun2 4173 . . . . 5 (⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩} β†’ ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
1412, 13ax-mp 5 . . . 4 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩})
15 eengv 28777 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) = ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
1614, 15eleqtrrid 2835 . . 3 (𝑁 ∈ β„• β†’ ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ (EEGβ€˜π‘))
17 fvex 6904 . . . . 5 (π”Όβ€˜π‘) ∈ V
1817difexi 5324 . . . . 5 ((π”Όβ€˜π‘) βˆ– {π‘₯}) ∈ V
1917, 18mpoex 8078 . . . 4 (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) ∈ V
2019a1i 11 . . 3 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) ∈ V)
211, 3, 10, 16, 20strfv2d 17162 . 2 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) = (LineGβ€˜(EEGβ€˜π‘)))
22 eengbas 28779 . . . 4 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
23 elntg.1 . . . 4 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
2422, 23eqtr4di 2785 . . 3 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = 𝑃)
2524difeq1d 4117 . . . 4 (𝑁 ∈ β„• β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2625adantr 480 . . 3 ((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2724adantr 480 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ (π”Όβ€˜π‘) = 𝑃)
28 simpll 766 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
29 elntg.2 . . . . . 6 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
30 simplrl 776 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
3128, 24syl 17 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (π”Όβ€˜π‘) = 𝑃)
3230, 31eleqtrd 2830 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ 𝑃)
33 simplrr 777 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))
3433eldifad 3956 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
3534, 31eleqtrd 2830 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ 𝑃)
36 simpr 484 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
3736, 31eleqtrd 2830 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ 𝑃)
3828, 23, 29, 32, 35, 37ebtwntg 28780 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ↔ 𝑧 ∈ (π‘₯𝐼𝑦)))
3928, 23, 29, 37, 35, 32ebtwntg 28780 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ↔ π‘₯ ∈ (𝑧𝐼𝑦)))
4028, 23, 29, 32, 37, 35ebtwntg 28780 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (𝑦 Btwn ⟨π‘₯, π‘§βŸ© ↔ 𝑦 ∈ (π‘₯𝐼𝑧)))
4138, 39, 403orbi123d 1432 . . . 4 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ ((𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©) ↔ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))
4227, 41rabeqbidva 3443 . . 3 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
4324, 26, 42mpoeq123dva 7488 . 2 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
4421, 43eqtr3d 2769 1 (𝑁 ∈ β„• β†’ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   βˆ– cdif 3941   βˆͺ cun 3942  βˆ…c0 4318  {csn 4624  {cpr 4626  βŸ¨cop 4630   class class class wbr 5142  β—‘ccnv 5671  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1c1 11131   βˆ’ cmin 11466  β„•cn 12234  2c2 12289  7c7 12294  cdc 12699  ...cfz 13508  β†‘cexp 14050  Ξ£csu 15656   Struct cstr 17106  ndxcnx 17153  Basecbs 17171  distcds 17233  Itvcitv 28224  LineGclng 28225  π”Όcee 28686   Btwn cbtwn 28687  EEGceeng 28775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-seq 13991  df-sum 15657  df-struct 17107  df-slot 17142  df-ndx 17154  df-base 17172  df-ds 17246  df-itv 28226  df-lng 28227  df-eeng 28776
This theorem is referenced by:  elntg2  28783  eengtrkg  28784
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