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Theorem elntg 28239
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 27688 . . 3 LineG = Slot (LineGβ€˜ndx)
2 fvex 6904 . . . 4 (EEGβ€˜π‘) ∈ V
32a1i 11 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ V)
4 eengstr 28235 . . . . 5 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
5 structn0fun 17083 . . . . 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 17085 . . . . . 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 6570 . . . 4 (𝑁 ∈ β„• β†’ (Fun β—‘β—‘(EEGβ€˜π‘) ↔ Fun ((EEGβ€˜π‘) βˆ– {βˆ…})))
106, 9mpbird 256 . . 3 (𝑁 ∈ β„• β†’ Fun β—‘β—‘(EEGβ€˜π‘))
11 opex 5464 . . . . . 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 4767 . . . . 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 4177 . . . . 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 28234 . . . 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 2840 . . 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 5328 . . . . 5 ((π”Όβ€˜π‘) βˆ– {π‘₯}) ∈ V
1917, 18mpoex 8065 . . . 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 17134 . 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 28236 . . . 4 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
23 elntg.1 . . . 4 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
2422, 23eqtr4di 2790 . . 3 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = 𝑃)
2524difeq1d 4121 . . . 4 (𝑁 ∈ β„• β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2625adantr 481 . . 3 ((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2724adantr 481 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ (π”Όβ€˜π‘) = 𝑃)
28 simpll 765 . . . . . 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 775 . . . . . . 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 2835 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ 𝑃)
33 simplrr 776 . . . . . . . 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 3960 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
3534, 31eleqtrd 2835 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ 𝑃)
36 simpr 485 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
3736, 31eleqtrd 2835 . . . . . 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 28237 . . . . 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 28237 . . . . 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 28237 . . . . 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 1435 . . . 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 3448 . . 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 7482 . 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 2774 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 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βˆͺ cun 3946  βˆ…c0 4322  {csn 4628  {cpr 4630  βŸ¨cop 4634   class class class wbr 5148  β—‘ccnv 5675  Fun wfun 6537  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  1c1 11110   βˆ’ cmin 11443  β„•cn 12211  2c2 12266  7c7 12271  cdc 12676  ...cfz 13483  β†‘cexp 14026  Ξ£csu 15631   Struct cstr 17078  ndxcnx 17125  Basecbs 17143  distcds 17205  Itvcitv 27681  LineGclng 27682  π”Όcee 28143   Btwn cbtwn 28144  EEGceeng 28232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-seq 13966  df-sum 15632  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-ds 17218  df-itv 27683  df-lng 27684  df-eeng 28233
This theorem is referenced by:  elntg2  28240  eengtrkg  28241
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